Math 252 Quiz 3 Solution
1. (10pts) Find the volume of the solid obtained by rotating the shaded region below
about the line y=−1. [Be sure to simplify your answer.]
Solution.
A cross section perpendicular to the x-axis at position x(0 ≤x≤1) is a washer with
inner radius x+ 1 and outer radius 2x+ 1 (see the picture above). Hence the area of
that cross section is
A(x) = π(2x+ 1)2−π(x+ 1)2=π(4x2+ 4x+ 1 −(x2+ 2x+ 1)) = π(3x2+ 2x)
Therefore the volume of the solid is
Z1
0
A(x)dx =πZ1
0
(3x2+ 2x)dx =π[x3+x2]1
0=π(1 + 1 −0−0) = 2π.