Math 241, Quiz 9. 10/31/12.
Name:
• Read problems carefully. Show all work. No notes, calculator, or text.
• There are 15 points total.
1. §15.3, #22 (7 points): Find the volume of the solid enclosed by the paraboloid z = x2 + 3y 2
and the planes x = 0, y = 1, y = x, and z = 0.
Solution: We have
1
Z
Z
1
2
Z
2
1
2
(x + 3y ) dy dx =
0
3
1
(x y + y )
x
y=x
1
=
3
2
(x + 1) − 2x3 dx
dx =
0
Z
1
Z
2
(−2x + x + 1) dx =
0
0
1
x3
− x4 +
+x
2
3
1
x=0
5
1 1
=− + +1= .
2 3
6
2. §15.3, #48 (8 points): Evaluate the double integral
Z 1Z 1
ex/y dy dx
0
x
by reversing the order of integration.
Solution: We compute
Z
1
Z
y
x/y
e
0
0
Z
dx dy =
1
x/y
y
ye
0
Z
dy =
x=0
0
1
1
(ye − y) dy = (e − 1)(y /2)
2
y=0
=
e−1
.
2