NAMES:
MATH 152
February 4, 2015
QUIZ 2
• 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) Find the volume of the solid formed when the region bounded by y = x3 , x = 0,
and y = 1 is revolved about the x−axis.
Solution: The larger radius from the x−axis is R(x) = 1 and the smaller radius from the
x−axis is r(x) = x3 . Thus the volume of the solid is
Z1
Z 1
6
2
2
π R(x) − r(x) dx = π
1 − x6 = π.
7
0
0
2. (3 points) Find the area bounded by the curves f = ex , g = e−x , x = 0 and x = 1.
Solution: On [0, 1], f > g so the area between the curves is
Z 1
1
1
ex − e−x dx = ex + e−x 0 = e + − 2.
e
0
NAMES:
MATH 152
February 4, 2015
3. (3 points) The base of a solid is the region bounded by y = 4 − x2 and y = 0. Find the
volume of the solid if cross sections perpendicular to the y−axis are squares.
Solution: The side of the square at x is 2x and so the area of the square as a function of y
is A(y) = 4(4 − y) = 16 − 4y. The volume of a thin slice is A(y) · dy = (16 − 4y) dy. The
volume is found by integrating the volume of a slice over the bounds 0 to 4:
Z4
Z4
Vslice (y) = A(y) dy
0
0
Z4
16 − 4y dy
=
0
4
= 16y − 2y 2
0
= 32.