MATH 101 V2A
February 4th – Practice problems
Hints and Solutions
Method of “washers”:
1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the
specified line.
(a) y 2 = x, x = 2y; about the y-axis.
Solution: We want to calculate the volume of the solid obtained by rotating the shaded region
around the y-axis.
y
2y = x
√
y= x
x
4
From this we can see that a washer will have inner radius r = y 2 and outer radius R = 2y. This
means that the volume of the solid is
Z 2
Z 2
4 3 1 5 2
64
2
2 2
2
4
V =π
(2y) − (y ) dy = π
(4y − y )dy = π
y − y =
π.
3
5
15
0
0
0
(b) y = x, y =
√
x; about x = 2.
Solution: We want to calculate the volume of the solid obtained by rotating the shaded region
about the line x = 2
y
=
x
√ x
=
y
y
x
1
2
From this we can see that a washer will have inner radius r = 2 − y and outer radius R = 2 − y 2 .
This means that the volume of the solid is
Z 1
Z 1
1 5
1
8
V =π
(2 − y 2 )2 − (2 − y)2 dy = π
(−5y 2 +y 4 +4y)dy = π − y 3 + y 5 + 2y 2 =
π.
3
5
15
0
0
0
2. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line.
(a) y = 0, y = sin(x) for 0 ≤ x ≤ π; about y = −2.
Z
Hint: Answer: V =
π
(2 + sin(x))2 − 22 dx.
0
(b) x2 − y 2 = 1, x = 3; about x = −2.
√
Z
Hint: Use the symmetry of the solid to simplify the integral. Answer: V = 2π
8
52 − (2 +
p
y 2 − 1)2 dy.
0
Method of cylindrical shells:
1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line.
(a) y = x2 , y = 0, x = 1, x = 2; about x = 1.
Solution: We want to calculate the volume of the solid obtained by rotating the shaded region
about the line x = 1.
2
y
y = x2
x
1
From this we can see that a cylindrical shell will have radius (x − 1) and height x2 . Therefore,
Z
2
2
1
√
2
3
x −x
(x − 1) · x dx = 2π
V = 2π
(b) y =
Z
2
dx = 2π
1
1 4 1 3 2
17
x − x =
π.
4
3 1
6
x − 1, y = 0, x = 5; about y = 3.
Solution: We want to calculate the volume of the solid obtained by rotating the shaded region
about the line y = 3.
y
3
y=
√
x−1
x
5
From this we can see that a cylindrical shell will have radius (3 − y) and height 5 − (y 2 + 1).
Therefore
Z 2
Z 2
1 4 2
2
2
3
3
2
V = 2π
(3−y)(5−(y +1))dy = 2π
12 − 3y − 4y + y dy = 2π 12y − y − 2y + y = 24π.
4 0
0
0
2. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line.
3
(a) y = log(x), y = 0, x = 2; about the y-axis.
Z
log(2)
y(2 − ey )dy.
Hint: Answer: V = 2π
0
(b) y =
1
1+x2 , y
= 0, x = 0, x = 2; about x = 2.
Z
2
(2 − x) ·
Hint: Answer: V = 2π
0
1
dx.
1 + x2
4