Calculus 1120, Review for Prelim 1 Dan Barbasch September 25, 2012 - - PowerPoint PPT Presentation

Calculus 1120, Review for Prelim 1

Dan Barbasch September 25, 2012

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 1 / 8

Fundamental Theorem of Calculus Practice Exercises chapter 5: any 43-72, 73-112, 121-128 Techniques of Integration

◮ Substitution ◮ Integration by parts ◮ Trigonometric Integrals

Practice Exercises chapter 5: Section 5.6: any 43-112 Practice Exercises chapter 8: any 1-8, 37-44 Applications

◮ Areas of regions between curves ◮ Volumes by slicing, disks/washers, shells ◮ Arclength ◮ Surface area ◮ Distance traveled, displacement, average value of a function

Section 5.4: 73-75 Section 5.5: 77, 78 Section 5.6: any 93 -110 Practice Exercises chapter 6: any 3, 4, 7-10, 13, 14, 19, 20, 22, 23

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 2 / 8

Fundamental Theorem of Calculus

F(x) = x

a

f (t) dt is an antiderivative of y = f (x); the unique one which satisfies F(a) = 0. The function f (x) must be continuous throughout an interval [a, b] containing x. b

a

f (x) dx = F(b) − F(a). A general formula (which you should not memorize but understand) d dx h(x)

g(x)

f (t) dt

• = f (h(x)) · h′(x) − f (g(x)) · g′(x).

Compute F ′(x) for F(x) = x3

x2

• 2 + sin2 t dt.

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 3 / 8

Techniques of Integration

Substitution. If you make a substitution u = h(x) you must be able to solve uniquely x = g(u), and then you substitute x = g(u) in the function, and dx = g′(u)du. Usually one takes short cuts. For definite integrals you must either carry the x and substitute at the end, or else change the limits of integration. Compute

• ex sec2(ex − 7) dx.

Integration by Parts.

• udv = uv −
• vdu.

Trigonometric Integrals. You need to know which trigonometric formulas to use. Compute

• sin 3θ cos 2θ dθ.

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 4 / 8

Area

A = b

a

|f (x) − g(x)| dx. You need to divide up the interval [a, b] into pieces where f or g is the larger of the two functions.

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 5 / 8

Volumes

Slices. V = b

a

A(x) dx. You need to choose an axis, say x, and then compute the area A(x) of the cross section at x. Disks/Washers. A special case of the method of slices; the cross sections are washers A(x) = π[r2(x)2 − r1(x)2]. V = b

a

π[r2(x)2 − r1(x)2] dx. r2, r1 need to be positive. The integrals may be in y depending on the axis of rotation. Shells. V = b

a

2πr(x)h(x) dx. h, r need to be positive.

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 6 / 8

Volumes

1 Sketch the region. 2 Choose a variable for integration. 3 Draw a slice perpendicular to the variable, and rotate to decide

washers/shells.

4 Write the general formula with the endpoints for the variable. 5 Compute r1, r2 or r, h in terms of the chosen variable. Label them. 6 Compute the integral.

6 (page 414). Find the volume of the solid with base the region bounded by y2 = 4x, and x = 1 with coross section perpendicular to the x−axis equilateral triangles with one edge in the xy−plane. 7 (page 414). Find the volume of the solid generated by revolving the region bounded by the x−axis, y = 3x4 and the lines x = 1 and x = −1 revolved by the (a) x− axis (b) y−axis (c) x = 1 (d) y = 3

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 6 / 8

Arclength/Surface Area

L = b

a

ds, A = 2π b

a

rds. ds =

• 1 + f ′(x)2 dx

ds =

• 1 + g′(y)2 dy

ds =

• x′(t)2 + y′(t)2 dt

The first two are special cases of the third. Using parametric forms for curves are very convenient. Example: length of the ellipse x2

a2 + y2 b2 = 1.

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 7 / 8

Other Applications

Motion: On a line: ds dt = v(t), dv dt = a(t), F = ma. Displacement is s(b) − s(a) = b

a

v(t) dt. Distance travelled is b

a

|v(t)| dt. Along a curve: Distance travelled is b

a

• x′(t)2 + y′(t)2 dt. If the motion does not go

back on itself this is the same as arclength. Average Value of a Function: 1 b − a b

a

f (x) dx.

Dan Barbasch Calculus 1120, Review for Prelim 1 September 25, 2012 8 / 8