MATH 1210-1: Quiz 11A
Solution
Please show your work and MARK your answer.
1. (5 points) The region bounded by y = 2x, the x-axis and x = 4 is revolved
about the y-axis. Find its volume.
Solution: By sketching the graph, we have a = 0, b = 4, f (x) = 2x and
g(x) = 0 in the formula of shell. Therefore,
Z
b
V = 2π
x(f (x) − g(x)) dx
a
Z
4
= 2π
x(2x − 0) dx
0
= 2π ·
¯4
2 3 ¯¯
x ¯
3 0
256
π
3
=
2. (5 points) The plane curve is given by x = 3 cos t, y = 3 sin t where
0 6 t 6 2π. Find the length of this curve.
Solution: We have a = 0, b = 2π, f (t) = 3 cos t and g(t) = 3 sin t in the
formula of length. Therefore,
Z
b
p
(f 0 (t))2 + (g 0 (t))2 dx
L=
a
Z
2π
p
(−3 sin t)2 + (3 cos t)2 dx
2π
√
=
0
Z
=
0
2π
= [3t]0
= 6π
9 dx
MATH 1210-1: Quiz 11B
Solution
Please show your work and MARK your answer.
1. (5 points) The region bounded by y = 3x, the x-axis and x = 6 is revolved
about the y-axis. Find its volume.
Solution:By sketching the graph, we have a = 0, b = 6, f (x) = 3x and
g(x) = 0 in the formula of shell. Therefore,
Z
b
V = 2π
x(f (x) − g(x)) dx
a
Z
6
= 2π
x(3x − 0) dx
0
¯6
= 2π · x3 ¯0
= 432π
2. (5 points) The plane curve is given by x = 2 cos t, y = 2 sin t where
0 6 t 6 2π. Find the length of this curve.
Solution: We have a = 0, b = 2π, f (t) = 2 cos t and g(t) = 2 sin t in the
formula of length. Therefore,
Z
b
p
(f 0 (t))2 + (g 0 (t))2 dx
L=
a
Z
2π
p
(−2 sin t)2 + (2 cos t)2 dx
2π
√
=
0
Z
=
0
2π
= [2t]0
= 4π
4 dx