Math 1210
Quiz 10
April 11th, 2014
Answer the following 2 questions. The value of every question is indicated at
the beginning of it. You may use scratch paper, but you can only turn in this
sheet. Please write your answer in the space provided. You have 20 minutes.
Name:
UID:
1. (14 points) Let R be the region in the first quadrant bounded by the graphs of y =
x = 4 and y = 0.
√
x,
(i) Write down the integral that would compute the volume of the solid obtained by
revolving R about the x-axis.
(a) (5 points) Using shells:
Solution:
2
Z
2πy(4 − y 2 ) dy
V =
0
(b) (5 points) Using disks:
Solution:
Z
4
π
V =
√ 2
x dx
0
(ii) (4 points) Use any method to write down the integral that would compute the
volume of the solid obtained by revolving R about the line x = −1.
Solution: Using disks we get
Z
4
V =
√
π ( x + 1)2 − 12 dx
0
Using shells the integral would be
Z 2
V =
2π(y + 1)(4 − y 2 ) dy
0
2. (16 points) Let R be the region in the first quadrant bounded by the graphs of y = x2
and y = 2 − x2 . Write down the integral that would compute the volume of the solid
obtained by revolving R about the y-axis.
(i) (8 points) Using shells:
Solution:
Z
V =
1
4πx(1 − x2 ) dx
0
(ii) (8 points) Using disks:
Solution:
Z
1
Z
1
0
Page 2
2
π(2 − y) dy
πy dy +
V =