Calculus 2

Math 131 Fall 2021 Exam 2

Name

Make sure to show all work to receive full credit for each question. Answers without supporting work will receive a score of zero. Also, please box your final answers, do your best to write clearly and legibly, and write your name on every odd-numbered page.

Multiple Choice

1. (2 pts each) Circle the formula for the volume obtained by rotating the illustrated region R about the axis x = k or y = k, as indicated.

(a) V = ∫ b a π((f(x) − k)2 − (g(x) − k)2) dx

(b) V = ∫ b a π((f(x) − g(x))2 − k) dx

(c) V = ∫ b a 2π(x − k)(f(x) − g(x)) dx

(d) V = ∫ d c 2π(k − y)(f(y) − g(y)) dy

(a) V = ∫ b a π(f(x) − k)2 dx

(b) V = ∫ b a π(k − f(x))2 dx

(c) V = ∫ b a 2π(k − x)f(x) dx

(d) V = ∫ b a 2π(x − k)f(x) dx

/ 4 pts

1

(a) V = ∫ b a 2π(k − x)(f(x) − g(x)) dx

(b) V = ∫ b a 2π(x − k)(f(x) − g(x)) dx

(c) V = ∫ b a 2π(k − x)(g(x) − f(x)) dx

(d) V = ∫ b a 2π(x − k)(g(x) − f(x)) dx

(a) V = ∫ d c 2π(y − k)(f(y) − a) dy

(b) V = ∫ d c 2π(y − k)(a − f(y)) dy

(c) V = ∫ d c 2π(k − y)(f(y) − a) dy

(d) V = ∫ d c 2π(y − k)(a − f(y)) dy

/ 4 pts

2

2. (1.5 pt each) Match the graph of the unbounded region R with the improper integral that evaluates its area. Solid black lines are boundaries; dotted black lines are asymptotes of f.

(a) (b)

(c) (d)

lim t→−∞

∫ b t

f(x) dx lim t→∞

∫ t a

f(x) dx

lim t→a+

∫ b t

f(x) dx lim t→b−

∫ t a

f(x) dx

/ 6 pts

3

Area of a Region

Set up an integral to find the requested area. Label each curve. Find and label the coordinates of the points of intersection. Make sure to show work justifying your choice of limits of integration. Do not evaluate the integral.

3. (5 pts) The area of R illustrated below, the region enclosed by x = (y−1)2 and y = x+1.

/ 5 pts

4

Volume of Surface of Rotation

4. Consider the region R bounded by the curves

x + y = 4 x = y2 − 4y + 4

(a) (4 pts) Graph the region. Label the curves. Find and label the points of inter- section. Show your work.

(b) Next, set up two integrals to find the volume obtained by rotating the region about the given axis. You may choose either the disc or cylinder method.

i. (5 pts) About the x-axis

ii. (5 pts) About the y-axis

/ 14 pts

5

Improper Integrals

5. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. Show your work

(a) (5 pts) ∫ 3 −2

1

x4 dx

(b) (5 pts) ∫ ∞ 1

e− 1 x

x2 dx

/ 10 pts

6