LAMC
Yun
Final A
Math 262
12/16/2022
Last name: _______________________________ First name: ____________________________________
Directions: Show all work and box your final answers. No work = No point
1. (6 pts) Evaluate

A sin B dx where A = x and B = 2 x .
2. (6 pts) Find the volume of the solid generated by revolving the region R bounded by given equation y ,
x -axis, y -axis, and x = ln 2 about the x -axis. (Find the exact answer. Do not estimate the
answer using your calculator.)
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A
 B dx where
3. (8 pts) Evaluate
A = 2e2 x and B = e2 x + 14ex + 48 . Be sure to show all your steps clearly.
4. (6 pts) Find the exact arc length of the following curve on the given interval.
(x
y=
2
+ 2)
3/2
3
on 0,1
2
5. (6 pts) Evaluate
A
1
B
dx where A = x2 and B = 25 + x2 . Which integration techniques did you
use? Please explain.
6. (8 pts) Determine whether the integral is convergent or divergent. Evaluate if it is convergent.
b
 f ( x) g ( x)dx where a = − , b = 0 ,
f ( x) = x , and g ( x) = e x
a
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7. (8 pts) A vertical cylindrical tank with diameter 2 m and height 5 meter is full of water. How much work
is required to pump all the water to the top of the tank? Use 1000 kg / m3 for the density of
water and 9.8 m /s2 for the acceleration due to gravity. (Volume of a cylinder: v =  r 2 h )
8. (6 pts) Given r = 1 − sin 
1  

2 6
Find the slope of the line tangent to the given polar curve at  ,
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9. (8 pts) Find the area of the shaded region.
10. (8 pts) Determine whether the series converges or diverges. State your reasons clearly and state the
test used.

k3
(a)  4
k =1 2k − 1

(b)
1
 n(ln n)
n=2
2
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11. (8 pts) (a) Write the power series using summation notation.
x−
x3 x5 x 7 x9
+ − + − ...
3! 5! 7! 9!
(b) Determine the radius of convergence.
1
as the sum of a power series.
1+ x

1
=  x n = 1 + x + x 2 + x3 + ... for x  1
*Hint:
1 − x n =0
12. (8 pts) (a) Express
(b) Find a power series representation for ln(1 + x) using your answer in (a).
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(c) Find a power series representation for x ln(1 + x) using your answer in (b).
6
13. (8 pts) Let f ( x) = x .
(a) Find a Taylor polynomial with degree 3 centered at a = 16 .
(b) Use your answer in (a) to approximate
18 . Round your answer to six decimal places.
14. (6 pts) Solve the initial value problem for a = 0 and b = 1 .
3 y 2 x2 + 1
dy
= − x, y ( a ) = b
dx
This is the end of the final exam.
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*Submit your file in Canvas Assignment Final Exam Submission. Thank you.
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