NAMES:
MATH 152
March 11, 2015
QUIZ 6
• Show all your work and indicate your final answer clearly. You will be graded not merely
on the final answer, but also on the work leading up to it.
1. (3 points) Set up, but do not evaluate, an integral for the length of the curve parameterized
by
x = et cos t, y = et sin t, 0 ≤ t ≤ π.
Solution: The length of a parametric curve is
s
Z π 2 2
dy
dy
L=
+
dt
dt
dt
0
Z πp
(−et sin t + et cos t)2 + (et cos t + et sin t)2 dt
=
0
2. (3 points) Find the exact area of the surface obtained by rotating the curve about the x-axis:
y = x3 , for 0 ≤ x ≤ 1.
Solution: The surface area is found using the formula found in the text:
Z 1
A = 2π
r·h
0
s
2
Z 1
dy
= 2π
y(x) 1 +
dx
dx
0
Z 1 p
= 2π
x3 1 + (3x2 )2 dx
0
Z 1 √
= 2π
x3 1 + 9x4 dx
0
Z 10
√
2π
=
u du (making the substitution u = 1 + 9x4 )
36 1
10 !
π 2 3 =
u2
18 3 1
π 3
=
10 2 − 1
27
NAMES:
MATH 152
3. (3 points) Find the arc length of the curve y = 12 ex + 12 e−x for 0 ≤ x ≤ 1.
Solution: This is just an application of the formula:
s
2
Z 1
dy
L=
1+
dx
dx
0
s
2
Z 1
1 x 1 −x
1+
e − e
=
2
2
0
Z 1r
1
1 1
=
1 + e2x − + e−2x
4
2 4
0
Z 1r
1 2x 1 1 −2x
e + + e
=
4
2 4
0
s
2
Z 1 1 x 1 −x
e + e
=
2
2
0
Z 1
1 x 1 −x
=
e + e
2
0 2
1
ex e−x =
−
2
2 0
=
e
1
−
.
2 2e
March 11, 2015