Calculus 2 test

Math3B Final Exam

Spring 2021 Online (200 Points)

Be sure to show all your work on the exam! Clearly indicate your solution to the problem by circling or boxing your answer. No credit will be given if you do not show how you reached your solution!

1. (15 Points) Given the graph below, sketch the polar curve

4

3

2

1

0

-1

-2

-3

-4

•

•

•

•

•

•

•

•

•

•

• •

2 π

π 2 3π

π2

2. (20 Points) Given the graph below, sketch the ellipse on the polar grid. Find the equation of the

ellipse in rectangular coordinates, the rectangular coordinates of the foci and the value of the

eccentricity. a c

e =

1

0

-1

-2

-3

-4

-5

-6

Equation: Foci: Eccentricity:

Solution Box

• •

•

• •

2 5 7 4 3 5 11 13 7 5 2

2 3 6 6 3 2 3 6 6 3 2 π π π π π π π π π π π

π π

3. (30 Points) Draw the two polar curves and find the area outside the circle but inside all four petals of

the rose curve. Hint: Use symmetry so that you only need one integral.

You must show your work to receive credit for this problem.

1 2r = 2 4 cos 2θr =

Next Page intentionally left blank for continued work for Problem #3.

4. (15 Points) Find a closed form for the nth partial sum ( ns ) of the series below. Then find the tenth partial

sum 10s using your closed form formula. Write you answer as a single fraction. Show your work!

2 4

1 3 2k k k

∞

=

   − + 

∑

5. (15 Points) Given 2x t= and 3 3y t t= − , answer the following questions. Show your work!

a) Find dy dx

.

b) Find the values of t that make your dy dx

equal to zero.

6. (20 Points)

a) 3y = : Washer Method

b) 3x = : Shell Method

(10 Points) The region shown in the figure to the right is bounded by the graphs;

ln , , and 0y x x e x= = = Setup the integral for the volume V of the solid obtained by revolving the region about the stated axis using the specified method. (DO NOT INTEGRATE!)

x e=

x axis−

y axis−

lny x=

0

7. (30 Points) Integrate. Show your work!

a) ( ) 0

3 1

2 3 dx

x−∞    

−   ∫ b)

4 1

lnx x dx ⋅ ∫

8. (20 Points) Integrate using trigonometric substitution.

2 2

1 4 9

dx x x −

∫

9. (20 Points) Use the Root Test to determine if the series converges or diverges.

( ) 3

1

2cos k

k

k

∞

= ∑

10. (20 Points) The tank below is full of heating oil with density 10 lb/ft3. Find the work required to pump all

the oil out the outlet.

4 ft

9 ft

2 ft

20 ft

11. (15 Points) A force of 10N is required to hold a spring that has been stretched 0.05 m beyond its natural

length. It the springs natural length is 0.10 m, how much work is required to stretch the spring from

0.15 m to 0.20 m?