AP Calculus AB Integration 2015-11-24 www.njctl.org Slide 3 / 175 - - PDF document

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AP Calculus AB

Integration

2015-11-24 www.njctl.org

Slide 2 / 175 Table of Contents

click on the topic to go to that section

Area Under a Curve (The Definite Integral) Trapezoid Approximation Riemann Sums Fundamental Theorem of Calculus, part I Indefinite Integrals U-Substitution Antiderivatives & The Fundamental Theorem of Calculus, part II Average Value & Mean Value Theorem for Integrals

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Mathematicians spent a lot of time working with the topic of derivatives, describing how functions change at any given instant. They then sought a way to describe how those changes accumulate

• ver time, leading them to discover the calculation for area under a
• curve. This is known as integration, the second main branch of

calculus. Finally, Liebniz and Newton discovered the connection between differentiation and integration, known as the Fundamental Theorem

• f Calculus, an incredible contribution to the understanding of

mathematics.

Integration Slide 4 / 175

Riemann Sums

Return to Table of Contents

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Formulas for the area of polygons, such as squares, rectangles, triangles and trapezoids were well known in many early civilizations. However, the problem of finding area for regions with curved boundaries (circles, for example) caused difficulties for early

• mathematicians. The Greek mathematician Archimedes proposed to

calculate the area of a circle by finding the area of a polygon inscribed in the circle with the number of sides increased indefinitely.

Area of Curved Shapes Slide 6 / 175

Consider the following velocity graph: 30 mph v(t) (mph) t (hours) 5 hrs How far did the person drive?

Distance Using Graphs Slide 7 / 175

Consider the following velocity graph: 30 mph v(t) (mph) t (hours) 5 hrs How far did the person drive?

Distance Using Graphs

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Answer You can see that this number can be obtained if we calculate the area under the velocity graph. So, the area of the rectangle in this case represents the total distance traveled.

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50 mph 5 hrs 30 mph v(t) (mph) t (hours) However, objects seldom travel at a constant speed. The area under this graph is still equal to the distance traveled but we need more than just simple multiplication to find it.

Non-Constant Speed Slide 8 / 175

Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to function analysis and found an approach for approximating the total area underneath a curve by dividing the total area into a series of rectangles. So, the area under the curve would be the sum of areas of the

• rectangles. Later, we will discuss how close this approximation is, and if

there is any possibility to calculate the exact area underneath the curved boundary.

Georg Friedrich Riemann Slide 9 / 175

Note: When we use the language "area under the curve" we are referring to the area between the function and the x-axis.

Area Under the Curve Slide 10 / 175

Note: When we use the language "area under the curve" we are referring to the area between the function and the x-axis.

Area Under the Curve

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Teacher Notes

Let students discuss what the area would be if it wasn't bounded by the x-

• axis. Hopefully they will conclude that

the area would be infinite.

vs.

Area Finite Area

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Rectangular Approximation Method is a way to estimate area by drawing rectangles from the x-axis up to the curve. The question is: What part of the "top" of the rectangle should lie on the curve? Also, how many rectangles should be used? The middle (MRAM) The right hand corner (RRAM) The left hand corner (LRAM)

RAM - Rectangular Approximation Method Slide 11 / 175

Example: Approximate the area under the curve y = x

2 on [0,1] with

a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). Is this approximation an

• verestimate or an underestimate?

Explain.

Riemann Sums Slide 12 / 175 Slide 13 / 175

Example: Approximate the area under the curve y = x

2 on [0,1] with

a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain.

Riemann Sums

Finally, calculate the area using LRAM. Have students discuss why this is an under approximation.

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Example: Approximate the area under the curve y = x

2 on [0,1] with

a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain.

Riemann Sums

If we look at our graph, we can see that all of the rectangles are below

• ur curve. Therefore, this

approximation is an underestimate. We calculated the area using LRAM to be

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Is this approximation an

• verestimate or an underestimate?

Example: Approximate the area under the curve y = x

2 on [0,1] with

a Riemann sum using 4 sub-intervals (rectangles) and right endpoints (RRAM).

Riemann Sums Slide 16 / 175

Slide 16 (Answer) / 175

Example: Approximate the area under the curve y = x

2 on [0,1] with

a Riemann sum using 4 sub-intervals (rectangles) and midpoints (MRAM).

Riemann Sums Slide 17 / 175 Slide 17 (Answer) / 175

*NOTE: MRAM ≠ LRAM + RRAM

2

Riemann Sums Slide 18 / 175

*NOTE: MRAM ≠ LRAM + RRAM

2

Riemann Sums

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Teacher Notes Technically, for some functions, MRAM could be the average of LRAM and RRAM; however, for most functions given in practice and on the exam, this will not be the case. Have a discussion with students about why this can't always be true.

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1 When approximating the area under the curve y=3x+2

• n [1,4] using four rectangles, how wide should each

sub-interval be?

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2 Approximate the area under y=3x+2 on [1,4]

using six rectangles and LRAM.

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3 Find the area under the curve on [-3,2]

using five sub-intervals and RRAM.

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The following notation is used when discussing Riemann sums and approximating areas. Some questions will use this notation, so it is important to be familiar with the meaning of each symbol. Using the symbols above, can you create a mathematical relationship between all 4 of them?

Riemann Sum Notation Slide 24 / 175

The following notation is used when discussing Riemann sums and approximating areas. Some questions will use this notation, so it is important to be familiar with the meaning of each symbol. Using the symbols above, can you create a mathematical relationship between all 4 of them?

Riemann Sum Notation

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Teacher Notes

start of interval end of interval number of sub-intervals width of each interval

Some students may need guidance coming up with a relationship. The most common answer students will come up with is usually because they have been calculating the width in earlier questions.

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Sometimes, instead of being given an equation for f(x), data points from the curve will be presented in a table. Provided the necessary information is in the table, you are still able to approximate area.

Riemann Sums with Tables Slide 27 / 175

Example: Approximate the area under the curve, f(x), on [-2,4] using right endpoints and n=3.

Riemann Sums with Tables Slide 28 / 175 Slide 29 / 175

Note: When using tabular data for Riemann Sums, not all sub- intervals need to be of equal width. If the question does not specify , then you are able to use the data provided - just make sure to account for the varying width.

Riemann Sums with Tables Slide 30 / 175

Example: Using the subintervals in the table, approximate the area under using a left hand approximation.

Riemann Sums with Tables

Answer

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Example: Using the subintervals in the table, approximate the area under using a right hand approximation.

Riemann Sums with Tables

Answer

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8

A B C D E F G H I Approximate the area under the function, , based on the given table values. Use a right hand approximation and 4 equal sub-intervals.

Answer

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9 Approximate the area under the function, , based on

the given table values and intervals. Use a left hand approximation.

Answer

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Slide 36 / 175 Refresher on Summations: Slide 37 / 175 Refresher on Summations:

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Answer

Remind students how to calculate the summations, before the next slide where they will write their own to represent Riemann Sums.

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To represent Riemann Sums using sigma notation, we need to know the number of rectangles on the interval, and height of each

• rectangle. We will let represent each rectangle.

Example: Use sigma notation to represent the area under the curve

• f on using 4 equal subintervals and left endpoints.

Sigma Notation Slide 38 / 175 Slide 38 (Answer) / 175

12

A B C D E Which of the following represents the approximate area under the curve on using midpoints and 3 equal subintervals?

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12

A B C D E Which of the following represents the approximate area under the curve on using midpoints and 3 equal subintervals?

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Answer

C

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13

A B C D E Which of the following represents the approximate area under the curve on using right endpoints and 6 rectangles?

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13

A B C D E Which of the following represents the approximate area under the curve on using right endpoints and 6 rectangles?

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Answer

D

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Trapezoid Approximation

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Trapezoid Approximation

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Teacher Notes Students may have already brought up the idea of using different shapes to approximate area; if not, you can bring it up now and ask for ideas to gain a closer approximation rather than using rectangles.

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Example: Approximate the area under the curve y = x

2 on [0,1] with

using a trapezoidal approximation. Recall the area of a trapezoid:

Trapezoidal Approximation Slide 42 / 175

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F

• r

f u t u r e r e f e r e n c e ! We could make our approximation even closer if we used parabolas instead of lines as the tops of our intervals. This is called Simpson's Rule but this is not on the AP Calculus AB exam.

Simpson's Rule Slide 43 / 175

14 The area under the curve on [1,3] is approximated

with 5 equal subintervals and trapezoids. What is the height of each trapezoid?

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14 The area under the curve on [1,3] is approximated

with 5 equal subintervals and trapezoids. What is the height of each trapezoid?

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Answer

Students may have difficulty seeing the height of the trapezoid as the value, as typically height is viewed vertically. It may help students to turn their papers sideways to calculate height and base length.

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15 The area under the curve on [1,3] is approximated

with 5 equal subintervals and trapezoids. What is the area of the 5th trapezoid?

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15 The area under the curve on [1,3] is approximated

with 5 equal subintervals and trapezoids. What is the area of the 5th trapezoid?

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Answer

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16 The area under the curve on [1,3] is approximated

with 5 equal subintervals and trapezoids. What is the approximate area?

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16 The area under the curve on [1,3] is approximated

with 5 equal subintervals and trapezoids. What is the approximate area?

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Answer

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17 What is the approximate area under the curve

• n [0,9] using the given trapezoids?

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17 What is the approximate area under the curve

• n [0,9] using the given trapezoids?

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Answer Remind students that intervals don't always have to be equal widths.

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18 What is the approximate fuel consumed during the hour

long flight using the trapezoids and given intervals?

Time (minutes) Rate of Consumption (gal/min) 10 20 25 30 40 40 60 45

Answer

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Area Under a Curve (The Definite Integral)

Return to Table of Contents

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We have used rectangles and trapezoids to approximate the area under curves so far. What

• ther techniques could we apply to gain a more

accurate approximation of the area?

What Do You Think? Slide 50 / 175

We have used rectangles and trapezoids to approximate the area under curves so far. What

• ther techniques could we apply to gain a more

accurate approximation of the area?

What Do You Think?

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Teacher Notes Students may bring up various ideas about using different shapes. Guide conversation to the idea that using more and more rectangles will produce a more accurate approximation.

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If n is the number of rectangles used, and we allow that number to approach infinity, the width of each rectangle, or , will become infinitely small, which we denote . While the Greek symbol Sigma is a capital S for "sum", the German mathematician, Liebniz, chose to use the elongated "S" symbol for integrals in 1675, and it is still the symbol we use today.

Infinite Rectangles Slide 51 / 175

lower limit of integration upper limit of integration integrand integral sign (infinitely small ) (the function being integrated) differential

The Definite Integral Slide 52 / 175

lower limit of integration upper limit of integration integrand integral sign (infinitely small ) (the function being integrated) differential

The Definite Integral

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Teacher Notes The expression is read as: "The integral from a to b of f(x) dx."

• r

"The integral from a to b of f(x) with respect to x."

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If is continuous on [a,b] then the area under the curve is the integral of from a to b. Note: The integral represents the "net area" meaning all area above the x-axis minus any area below the x-axis.

The Definite Integral Slide 53 / 175

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19 Given the following:

Find the value of the following integral, if possible.

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19 Given the following:

Find the value of the following integral, if possible.

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Answer

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20 Given the following:

Find the value of the following integral, if possible.

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20 Given the following:

Find the value of the following integral, if possible.

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Answer

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21 Given the following:

Find the value of the following integral, if possible.

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21 Given the following:

Find the value of the following integral, if possible.

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Answer

Not possible; not given enough information

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22 Given the following:

Find the value of the following integral, if possible.

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22 Given the following:

Find the value of the following integral, if possible.

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Answer

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23 Given the following:

Find the value of the following integral, if possible.

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23 Given the following:

Find the value of the following integral, if possible.

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Answer

Not possible; not given enough information

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24 Given the following:

Find the value of the following integral, if possible.

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24 Given the following:

Find the value of the following integral, if possible.

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Answer

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25 Given the following:

Find the value of the following integral, if possible.

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25 Given the following:

Find the value of the following integral, if possible.

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Answer

Not possible; not enough information given

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26 Given the following:

Find the value of the following integral, if possible.

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26 Given the following:

Find the value of the following integral, if possible.

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Answer

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Example: Using your knowledge of geometry, evaluate the following integral:

Evaluating Integrals Using Geometry Slide 63 / 175

Example: Using your knowledge of geometry, evaluate the following integral:

Evaluating Integrals Using Geometry

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Answer

Students should recognize the equation for the semicircle. At this point, they will simply use the Area formula to evaluate the integral.

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27 Evaluate:

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27 Evaluate:

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Answer

Again, at this point students are simply using the geometric representation to calculate the area, as they haven't encountered antiderivatives yet.

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28 Given the fact that use your knowledge of

trig functions to evaluate:

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28 Given the fact that use your knowledge of

trig functions to evaluate:

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Answer

Slide 65 (Answer) / 175 DISCUSSION:

What does it mean when the area under the curve on a given interval equals zero?

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DISCUSSION:

What does it mean when the area under the curve on a given interval equals zero?

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Teacher Notes

There is equal area above and below the x-axis on the given interval.

Slide 66 (Answer) / 175

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Antiderivatives & The Fundamental Theorem

• f Calculus Part II

Slide 67 / 175

In previous examples, we have either known the shape of the function to calculate the area, or information about the area was given to us. Now we will discover how to calculate the integral (area) for almost any function.

What about other functions? Slide 68 / 175

Let's take a look back at the example we did in the previous section: Let's imagine this representation is somebody running 5mph from 1

• 'clock to 7 o'clock, it's simple to see the person traveled 30 miles. The

area under the velocity function gives us the distance traveled. It was this notion that allowed mathematicians to discover the relationship between a function and it's derivative, and furthermore, a function's antiderivative.

Recall: Slide 69 / 175

So, what exactly is an antiderivative?

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So, what exactly is an antiderivative?

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Teacher Notes

Allow students to discuss their ideas of an antiderivative and record all of their

• comments. Some common phrases/words

may include:

"undo derivative" "reverse" "go backwards" "find original function"

Slide 70 (Answer) / 175

Antiderivative: Slide 71 / 175 Slide 71 (Answer) / 175 Slide 72 / 175

Slide 72 (Answer) / 175

Example: Evaluate the following integral:

Fundamental Theorem of Calculus, Part II Slide 73 / 175 Slide 73 (Answer) / 175

Calculating Antiderivatives Slide 74 / 175 Calculating Antiderivatives

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Teacher Notes

This slide is meant to help them with antiderivatives, but needs to be mathematically accurate as well. Since we are not delving into indefinite integrals quite yet, you may need to briefly discuss with students why there are no bounds for integration and why the +C constant value is written with the antiderivative. They will be introduced to this with indefinite integrals and will even revisit definite integrals again to see why it wasn't needed.

Slide 74 (Answer) / 175

One thing to keep in mind is that it does not matter what variables are represented in your integral, as long as they match the variable you are integrating with respect to.

Fundamental Theorem of Calculus, Part II Slide 75 / 175

29 Evaluate:

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29 Evaluate:

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Answer

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30 Evaluate:

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30 Evaluate:

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Answer

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31 Evaluate:

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31 Evaluate:

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Answer

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32 Evaluate:

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32 Evaluate:

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Answer

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34

A B C D E F Evaluate:

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34

A B C D E F Evaluate:

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Answer

C

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The Fundamental Theorem of Calculus, Part I

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Most mathematicians consider The Fundamental Theorem of Calculus as the most important discovery in the history of

• mathematics. This relationship between differentiation and

integration provided a critical connection between the two fields which first appeared unrelated.

Fundamental Theorem of Calculus Slide 83 / 175 Slide 84 / 175

Slide 84 (Answer) / 175

If , then using our previous knowledge of integration, we can evaluate :

Fundamental Theorem of Calculus, Part I Slide 85 / 175 Slide 85 (Answer) / 175

Now, taking this one step further... Let's calculate the derivative of f(x).

Fundamental Theorem of Calculus, Part I Slide 86 / 175

Putting it all together, we calculated the following: Can you make any observations about methods to get from the first equation to the last and omitting the middle step?

Fundamental Theorem of Calculus, Part I Slide 87 / 175

Putting it all together, we calculated the following: Can you make any observations about methods to get from the first equation to the last and omitting the middle step?

Fundamental Theorem of Calculus, Part I

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Teacher Notes Have students discuss their thoughts and ideas about this

• process. It is not always clear to

all students straight away, so ask leading questions about what the function and derivative have in common, etc.

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Fundamental Theorem of Calculus, Part I

If is a continuous function on , then Our work on previous slides has led us to the discovery of the Fundamental Theorem of Calculus, Part I which states:

Slide 88 / 175 Fundamental Theorem of Calculus, Part I

In common terms, if taking the derivative of an integral, evaluated from a constant to x, you can simply replace the variable in the integral with x for your derivative. Note: This only applies when the lower limit of integration is constant and the upper limit is x. We will soon discuss how to evaluate if it is something other than x.

Slide 89 / 175 FTC (Part I)

Let's look at how this theorem works with another function: Find the derivative of:

Slide 90 / 175

FTC (Part I)

Let's look at how this theorem works with another function: Find the derivative of:

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Teacher Notes It is worth mentioning to students that an added benefit

• f using the FTC part I is that it

allows us to differentiate integrals for which we do not know the antiderivative of the integrand.

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Example: Given Find .

FTC (Part I) Slide 91 / 175

Example: Given Find .

FTC (Part I)

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Answer

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Example: Given Find .

FTC (Part I) Slide 92 / 175

Example: Given Find .

FTC (Part I)

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Answer

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Now let's discuss special circumstances of the FTC: What do you notice is different about the following example? Given Find .

Special Circumstances Slide 93 / 175

Now let's discuss special circumstances of the FTC: What do you notice is different about the following example? Given Find .

Special Circumstances

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Answer

Students should recognize the bounds of integration are flipped. They can use the properties of integrals to rewrite the integral and apply the FTC.

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Another special circumstance of the FTC: What do you notice is different about the following example? Given Find .

Special Circumstances Slide 94 / 175

Another special circumstance of the FTC: What do you notice is different about the following example? Given Find .

Special Circumstances

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Answer

Students should recognize the upper limit

• f integration is not just x. We must apply

a form of the Chain Rule with the FTC. Extra step:

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One more special circumstance of the FTC: What do you notice is different about the following example? Given Find .

Special Circumstances Slide 95 / 175 Slide 95 (Answer) / 175

35

A B C D E Find

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35

A B C D E Find

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Answer

C

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36

A B C D E Find

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36

A B C D E Find

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Answer

B

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37

A B C D E Find

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37

A B C D E Find

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Answer

D

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38

*From the 1976 AP Calculus AB Exam If , find

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38

*From the 1976 AP Calculus AB Exam If , find

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Answer

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39

A B C D E Find

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39

A B C D E Find

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Answer

A

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41

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41

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Answer

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Average Value & Mean Value Theorem for Integrals

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50 mph 5 hrs 30 mph v(t) (mph) t (hours) Recall the graph below which we used at the beginning of the unit regarding a non-constant speed. This section will allow us to calculate the average value (in this case, the average velocity) on a given

• interval. If we are given a function, we can then apply our knowledge of

integrals to calculate this value.

Average Value Slide 105 / 175 Average Value

The Average Value of f(x) is the height of the rectangle with base (b-a) and whose area equals the area under the graph of f(x) between x=a and x=b. Imagine all of the area under the given curve is transformed into a

• rectangle. This rectangle has the

same base length as the interval. The height is the average value the functions takes on within that

• interval. The star represents the

average value.

Slide 106 / 175 Average Value

If is a continuous function on . the average value of the function is:

Slide 107 / 175

Slide 108 / 175 Slide 108 (Answer) / 175 Average Value

A graphical representation of our answer from the previous example:

Slide 109 / 175

Average Value

Another example: Find the average value of the function over the given interval.

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Note: The average value of a function is not found by averaging the 2 y-values of the interval boundaries.

Average Value Slide 111 / 175

Note: The average value of a function is not found by averaging the 2 y-values of the interval boundaries.

Average Value

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Teacher Notes

This is an important idea to emphasize to students. Review the last example to make the point clear. It is a common misconception to just calculate f(0) and f(4) and average them, which in this case would equal

• 1. When applying the formula correctly

we found the average value to be 1.333.

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43 Find the average value of the function on the given interval.

A B C D E F

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43 Find the average value of the function on the given interval.

A B C D E F

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Answer

C

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Much like the MVT for Derivatives told us at which value, c, the slope was equal to the average slope; the Mean Value Theorem for Integrals will tell us at which value, c, the function reaches it's average value.

Mean Value Theorem for Integrals Slide 116 / 175

50 mph 5 hrs 30 mph v(t) (mph) t (hours)

Mean Value Theorem for Integrals

Let's just say we already calculated the average value for our function, and found it to be 30mph. The Mean Value Theorem for Integrals states that at at least one point, c, the function must take on it's average value. What does that mean for our example above?

Slide 117 / 175

50 mph 5 hrs 30 mph v(t) (mph) t (hours)

Mean Value Theorem for Integrals

Let's just say we already calculated the average value for our function, and found it to be 30mph. The Mean Value Theorem for Integrals states that at at least one point, c, the function must take on it's average value. What does that mean for our example above?

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Teacher Notes Students should discuss and conclude that although the car was not travelling at 30mph the entire trip, there must have been at least one point on the trip that it was traveling at that

• speed. In this case, by
• bserving the graph, 2 points.

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Mean Value Theorem for Integrals

If f(x) is a continuous function on [a,b], then at some point, c, where a<c<b

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45 Find the value(s) of c that satisfy the MVT for integrals.

A B C D E F

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45 Find the value(s) of c that satisfy the MVT for integrals.

A B C D E F

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Answer

E

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48 Find the value(s) of c that satisfy the Mean Value

Theorem for integrals.

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Indefinite Integrals

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Indefinite Integrals

So far, we have been calculating definite integrals, meaning our integral had bounds on both sides. Next, we will consider what happens when our integrals do not contain upper or lower limits, also known as indefinite integrals.

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Reflect

Talk, in teams, about what you noticed about the functions written in each box.

Slide 127 / 175 Indefinite Integrals = Antiderivatives

When we are given integrals without upper and lower limits, the question is really asking us to give the antiderivative. The only catch is that we MUST include the +C constant with each

• antiderivative. This represents that technically any constant could

be added to the original function. In the next unit, we will see that sometimes additional information (like an initial condition) is provided in order to find out exactly what the original function was. If not, it is imperative to include the +C.

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Example

Evaluate:

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Evaluate:

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Answer

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Evaluate:

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Example

Evaluate:

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Answer

Slide 131 (Answer) / 175 How Important is the Constant?

It might seem like a trivial piece, but when evaluating indefinite integrals, it is extremely critical to include the +C value. In fact, when questions arise on the free response portion of the AP Exam, 1 point is

• ften awarded just for including the +C. Keep in mind, only 9 points

area available on each free response question. Including the constant of integration not only demonstrates that you have a good understanding of the behavior of antiderivatives, but also allows you to take questions further as we will see with differential equations in the next unit.

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Slide 133 (Answer) / 175 Antiderivatives Involving Exponential and Natural Log Functions Slide 134 / 175 Antiderivatives Involving Exponential and Natural Log Functions

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Teacher Notes It's important to point out the necessity of the absolute value bars in ln|x|. No matter what number e is raised to, the result will be positive.

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Note: These response questions intentionally avoid using multiple choice to allow students the opportunity to remember to include the +C with their answer.

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Slide 141 (Answer) / 175 Recap: Definite vs. Indefinite Integrals

Turn to a partner to discuss the similarities and differences of definite and indefinite integrals.

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Return to Table of Contents

U-Substitution

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Recall when we were asked to find the derivative of a composite

• function. We had to utilize the Chain Rule to take the derivative

correctly. U-Substitution is a similar tool used to find the antiderivative of more complex functions, and essentially the "undo" of the Chain Rule.

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Let's start with an example. Evaluate: Notice, upon first glance this looks like a fairly complex integral. But, if we let We can then find the differential: This allows us to rewrite the integral in terms of u, to make the integration easier. Notice how much less intimidating this integral is!

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U-Substitution

Now, let's finish evaluating the integral. DON'T FORGET! Substitute your expression back in for u.

Slide 146 / 175 Steps for U-Substitution

• 1. Choose your value for u.
• 2. Find the differential (take derivative & solve for du).
• 3. Make the substitution into original problem.
• 4. Integrate as usual.
• 5. Substitute back in for u.

Slide 147 / 175 Deciding Values for U

In our previous example, the value for u was given; however, this won't always be the case. So, how do we decide or choose the correct value for u? The best advice is to look for an expression in the integral for which you also see that expression's derivative.

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Good choices for u usually include: · An expression raised to a power · An expression in a denominator · An expression as an "inside" function of a composition

More Tips for Choosing Your "U"

Avoid choosing a value for u which is too complex, at the same time make sure it's not too simple (i.e. don't let u=x)

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Consider the following example: If we let: Then, What do you notice?

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New Circumstance

Consider the following example: If we let: Then, What do you notice?

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Teacher Notes

Allow students to discuss their findings in pairs or groups. Students should recognize that the exact equation we have for du is not found in the original integral. It's different by a factor of 2.

Slide 155 (Answer) / 175 Fixing the Problem

Evaluate:

Slide 156 / 175 Fixing the Problem

Evaluate:

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Answer

In order to make our substitution this time, we must make one additional step. Divide by 2 on both sides: So, our new integral becomes:

(Recall the properties of integrals allow us to bring the constant outside the integral)

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One More Situation to Consider

Consider the following example:

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61 Evaluate using u-substitution (if needed):

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61 Evaluate using u-substitution (if needed):

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Make sure students understand the importance

• f adding the +C!!!

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A B C D E Evaluate using u-substitution (if needed):

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A B C D E Evaluate using u-substitution (if needed):

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A

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63 Evaluate using u-substitution (if needed):

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63 Evaluate using u-substitution (if needed):

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A B C D E Evaluate using u-substitution (if needed):

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A B C D E Evaluate using u-substitution (if needed):

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B

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A B C D E Evaluate using u-substitution (if needed):

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A B C D E Evaluate using u-substitution (if needed):

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By this point, students are in the habit of using u-substitution. However, this question can be approached by simply distributing first, and then integrating like normal.

D

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66 Evaluate using u-substitution (if needed):

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66 Evaluate using u-substitution (if needed):

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67 Evaluate using u-substitution (if needed): CHALLENGE

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Slide 164 (Answer) / 175 U-Substitution with Definite Integrals

Definite integrals may sometimes require u-substitution as well, but it is important to take extra caution when dealing with the limits of integration. Let's look at an example and then address the issue with the limits. Note: When we are given the original problem, all expressions and values are in terms of the variable x, including the bounds of integration.

Slide 165 / 175 U-Substitution with Definite Integrals

From this point, you have 2 options:

• 1. Integrate, substitute the

expression with x back in and use original bounds.

• 2. Change bounds in terms
• f u, integrate and use the

new bounds to evaluate.

OR

We will finish this example showing both methods to compare.

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U-Substitution with Definite Integrals

Option #1: Integrate, substitute the expression with x back in and use original bounds.

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Option #2: Change bounds in terms of u, integrate and use the new bounds to evaluate.

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A B C D Which values correspond to the correct bounds of integration in terms of u?

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A B C D Which values correspond to the correct bounds of integration in terms of u?

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A

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A B C D Which values correspond to the correct bounds of integration in terms of u?

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A B C D Which values correspond to the correct bounds of integration in terms of u?

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C

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A B C D Given that , which of the following answers is equivalent to ? E

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A B C D Given that , which of the following answers is equivalent to ? E

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D

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A B C D E

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B

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74 Evaluate:

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