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Math 1432 Test 2 Review (Covers Chapter 7) Solve the problems in this review sheet on notebook paper. Staple all pages together; write your name and PS ID on the cover page and turn it in to your TA in lab on the due date indicated on CASA. You will not get the review sheet back; you may want to make a copy before turning it in. Must show work!

1. Find the following integrals.

a. 4 cos

2 sinx x

e dxx

b. 2

sinh

(2 cosh )

xdx

x

c. 1 ln xe

dxx

d.

dxx

x

1

2

e. 2

4 6

4

xdx

x

f. 2

arctan( )

1

xdx

x

g.

2

3

021

1dx

x

h. 2

1

9 4dx

x

i. 2

5

36dx

x

j. xe

dxx

k. 1

2

0

8x x dx

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2. Find the average value of the following function over the given interval.

a. 3f x x x 0 2, ,

b. 2g x t 6t 1 2, , c. f x x 2x 0 4cos sin , , /

3. Sketch the region bounded between the graphs of f and g, and then find the area

of this region.

a. 2f x x 12 g x 4x,

b. 2f x 9 x g x 8x,

c. 3f x x g x 8x,

4. Consider the curves: 2x y 16 and x 10y .

Give an expression involving integrals that gives the area bounded by these curves.

5. Let R be the region bounded by 2f x x 2x and the x-axis. The region R is the base of a solid for which the cross sections perpendicular to x-axis are squares. Find the volume of that solid. Repeat this problem with cross sections that are semicircles.

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6. Consider the region bounded by 2f x x and g x 4x .

a. Calculate the volume of the solid formed if this region is rotated around the x-axis.

b. Calculate the volume of the solid formed if this region is rotated around the y-axis.

7. Consider the region bounded by 2x y and xy4

.

a. Write the integral that gives the volume if this region is rotated around the x-axis. Use Disk/washer method.

b. Write the integral that gives the volume if this region is rotated around the x-axis. Use Shell method.

c. Write the integral that gives the volume if this region is rotated around the y-axis. Use Disk/washer method.

d. Write the integral that gives the volume if this region is rotated around the y-axis. Use Shell method.

8. Consider the region in the first quadrant bounded by 3f x x and g x x .

a. Write the integral that gives the volume if this region is rotated around the x-axis.

b. Write the integral that gives the volume if this region is rotated around the y-axis.

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9. Find the arc length of the following curves:

a. 3 221 1 2

3/f x x , x ,

b. 42 0 2xf x e , x , (set up the formula only)

10. Let R be the region bounded by the x-axis and the curve 2 ln(10 )f x x x for 1 2x . Write a formula for the surface area of the solid generated when this region is rotated about the x-axis.

11. Sketch the region bounded by the x-axis and the curve 2f x x x .

a. Find the area of this region.

b. Find the centroid of this region.

c. Find the volume of the object formed when this region is rotated around the x-axis. (You can use Pappus’s theorem if you prefer.)

d. If this region is the base of a solid for which the cross-sections perpendicular to the x-axis are squares, find the volume of that solid.

12. If the centroid of a region is (10,12) and the area is 5, find the volume of the solid formed when this region is rotated about the x-axis. Assume that the region does not contain the x-axis.

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13. Give the general solution for

a. 5dy

ydx

.

b. xdyye

dx

c. 2dy

y sin xdx

d. 2 1

dy y

dx x

e. 24dy

y xdx

f. 2 9

dy y xy

dx y

g. 2y

sin xdy

dx e

14. Find the specific solution given the initial condition: 2 , (0) 6dy

y x ydx

.

15. The number N of bacteria in a culture is given by kteN 200 . If N = 300 when t =

4 hours, find k and then determine approximately how long it will take for the number of bacteria to triple in size.

16. The initial population of a bacteria culture is 1,000. If the population increases by

10% in 20 minutes, what is the doubling time? What is the population after 2 hours?

17. The initial mass of a substance is 10 grams. The mass decreases by 5% in one year. Find the general formula for the mass of this substance in terms of t (time).

18. Suppose that the population of Zeegers grows at a rate proportional to itself, doubling every 12500 years. When the Zeeger population has reached 93 percent

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more than their current population, they plan to invade Earth. How many years will it be before the Zeegers attack Earth?

19. At what rate r of continuous compounding does a sum of money double in 15 years?

20. Determine if each integral is improper. If it is improper, state why, re-write it using proper limit notation, and evaluate. If divergent, state “diverges”.

a. dx

b. dx

c.

d.

e. 0

1

1

ex dx

f.

4

1/31 2

1d

xx

g.

1

1 x2 dx

h. 2

5

(x 1)1/2dx

i. 31

1

( 4)dx

x

27

0

3/2x

4

0 4

1

x

9

2/3

1

1x dx

4

0

xedx

x