MTH 133 Final Exam December 9th, 2019

Standard Response Questions. Show all work to receive credit. Please BOX your final answer.

1. Let R be the region in the first quadrant bounded by y = sin x, y = cosx, and x = 0.

(a) (2 points) Sketch the region R on the axes to the right.

x

y

1

�1

(b) (3 points) Set up BUT DO NOT EVALUATEthe definite integral representing the volumeof the resulting solid if the region R is rotatedabout the x-axis.

2. (7 points) Evaluate the integral

Z 3

0

dx

9 + x2

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2 good

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3tanodxs

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MTH 133 Final Exam December 9th, 2019

3. Evaluate the following integrals.

(a) (6 points)

Zsec3 x tan x dx

(b) (6 points)

Zy · ln y dy

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us secx 3du secx faux

J n'du fu't C 13

f sedx 1C

us thy dus y dy 3dustydy v ty

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I y hy f y't C

In

fmr

MH

h

MTH 133 Final Exam December 9th, 2019

4. Determine whether each of the series below is convergent or divergent. For full credit, you must showyour work and indicate which test(s) you used.

(a) (6 points)1X

n=2

pn

ln(3n)

(b) (6 points)1X

n=2

2

npn+ 1

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for n 2

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I we know E n diverges by DCT

Use LCT with bn nth 1 2

I L air2 i if z 13

So since I come by p series

nmust also come by LCT

I

MELifEm

I

MTH 133 Final Exam December 9th, 2019

5. (6 points) Evaluate

Zsin x

xdx as an infinite series. Express your answer in sigma notation.

6. (6 points) Consider the curve C given by the parametric equations:

x = 2 + 3t, y = cosh(3t)

Find the arc length of the curve on the interval 0 t 1.

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sinx c is X JnIntl

as ItIs EC c n

X 3 y SinhGt 3

9t9sinh dtO

9cosh 3t dt

3coshGt Dt Sinh 3T

Sinh 3

ha

to

A

MTH 133 Solutions to Final Exam December 9th, 2019

7. Two curves in polar coordinates are given by equations r = 1 and r = 1− sin θ, respectively.(a) (3 points) Sketch both curves.

Solution: See here.

(b) (3 points) Determine the intersection pointsof these two curves. Express these points inpolar coordinates (r, θ).

Solution: The points of intersection occur at (1, 0) and (1, π).

(c) (6 points) Set up BUT DO NOT EVALUATE the definite integral representing the area that liesinside the circle r = 1 and outside the cardioid r = 1− sin θ.

Solution: ∫ π/2

0

(1− (1− sin θ)2) dθ

Page 5 of 10

Math 133 Multiple Choice - Correct Answers FS19

8. E

9. B

10. C

11. D

12. A

13. B, C

14. E

15. A

16. E

17. C

18. E

19. C

20. A

21. A

22. A

23. B

24. B