4/6/2010 Integrals Integrals MAC 2233 Antiderivatives A function - - PDF document

4/6/2010 1 Integrals Integrals

MAC 2233

A function F is an antiderivative of f on an interval I if _______________ for every x in I.

Antiderivatives

If G is an antiderivative of f, then every antiderivative of f must have the form where C is ___________ The process of finding all antiderivatives of a function is called antidifferentiation or integration

Indefinite Integrals

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Rules of Integration

, where , are constants k dx kx C k C = +

∫

1

∫

1

1 , where 1 1

n n

x dx x C n n

+

= + ≠ − +

∫

( ) ( ) cf x dx c f x dx =

∫ ∫

[ ( ) ( )] ( ) ( ) f x g x dx f x dx g x dx ± = ±

∫ ∫ ∫

Rules of Integration

1 ln | | dx x C x = +

∫

x x

e dx e C = +

∫

Example

• Integrate

4 dx −

∫

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Example

• Integrate

2

1 3 6 x x dx + −

∫

Example

• Integrate

1.7 2.5

x x dx

−

−

∫

Example

• Integrate
• Rewrite

1 3

1 2x dx x

−

+

∫

• Rewrite

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Example

• Integrate
• Rewrite

5

9 x dx

∫

• Rewrite

Example

• Integrate
• Rewrite

0.4 0.4

4.2 2 3

x

x e dx x + −

∫

• Rewrite

Homework

• p. 381 problems 1-29 odd, 37, 43, 45

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How do we integrate ? 1. Let u = g (x), where g is part of the integrand, usually the

• f the composite

S ubstitution

2 3 7

3 ( 1) x x dx +

∫

usually the _______________ of the composite function f (g (x)).

• 2. Compute ______________.
• 3. Use the substitution __________________ to

convert the entire integral into one involving only u.

• 4. Evaluate the resulting integral.
• 5. Replace u by g (x) to obtain the final solution as a

function of x.

Example

• Integrate

3 2 3 2

3 ( 2) t t dt +

∫

Example

• Integrate

2 3 2

3 2 ( 2 ) x dx x x + +

∫

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Example

• Integrate ∫

+ dx x x

7 4 3

) 9 (

3

2 x dx x +

∫

Example

• Integrate

1 7 5 dx x −

∫

Example

• Integrate

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2

x

xe dx

∫

Example

• Integrate

Example

The current circulation of the Investor’s Digest is 3000 copies per week. The managing editor of the weekly projects a growth rate of weekly projects a growth rate of copies per week, t weeks from now, for the next 3

• years. Based on her projection, what will the

circulation of the digest be 125 weeks from now?

From Calculus for the Managerial, Life, and Social Sciences, 6th ed. By Tan, 2003, example 12, p.406.

Homework

• p. 394 problems 3-35 odd, 45, 51, 55, 61, 67

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How do we calculate the area of the region bounded by the graph of a nonnegative function, f, the x-axis, and the vertical lines x = a and x = b?

Area Under the Curve

Let f be a nonnegative, continuous function on [a, b]. Then the area of the region under the graph of f is

Area under the curve

graph of f is Let f be a continuous function defined on [a, b]. If

The Definite Integral

exists for all choices of x1, … , xn in the subintervals

• f [a, b] then this limit is called the definite

integral of f from a to b and we write

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Properties of the Definite Integral

( ) ( )

b b a a

kf x dx k f x dx =

∫ ∫

[ ( ) ( )] ( ) ( )

b b b

f x g x dx f x dx g x dx ± ±

∫ ∫ ∫

[ ( ) ( )] ( ) ( )

a a a

f x g x dx f x dx g x dx ± = ±

∫ ∫ ∫

( ) ( ) ( )

b c b a a c

f x dx f x dx f x dx = +

∫ ∫ ∫

( )

a a

f x dx =

∫

Let f be a continuous function on [a, b]. Then

The Fundamental Theorem of Calculus

where F is any antiderivative of f ; that is F’(x) = f (x). We write

Example

• Find the area of the region under f (x) = 4x – 1
• n the interval [2, 4].

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Example

• Evaluate

1

4 x dx

−

−

∫

• Evaluate

Example

2 5 3 1

1 t t dt − +

∫

Example

• Evaluate

1 2 2

3 ( 1) x x dx −

∫

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Example

• Evaluate

3 4 4 1 (2

) t dt t

−

−

∫

Example

• Evaluate

1

1 4 5 dx x

−

−

∫

Net Change

• The definite integral represents the net change

in the antiderivative function

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Example

A certain oil well that yields 400 barrels of crude

• il a month will run dry in 2 years. The price of

crude oil is currently $95 per barrel and is crude oil is currently $95 per barrel and is expected to rise at a constant rate of 30 cents per barrel per month. If the oil is sold as soon as it is extracted from the ground, what will be the total future revenue from the well?

From Calculus for Business, Econom ics and the Social and Life Sciences, 10 th ed. By Hoffmann & Bradley, 2007, problem 50, p.412.

Homework

• p. 410 problems 1-29 odd, 41, 43, 45, 49, 53, 55,

57 59 63 57, 59, 63