Math 9B Worksheet 10
1. (Washer) Consider the region between y = x− 1 and y = 2√x− 1.
(a) Find the endpoints and shade the region on a plane.
(b) Consider the revolutionary solid about the x-axis. Set up the integral for that volume using the
washer method. Solve whichever you prefer.
(c) Consider the revolutionary solid about the axis x = 7. Set up an integral for that volume using
the washer method. Do not solve.
2. (a) Show the volume of a right cone of height h and radius r is 13πr
2h using solids of revolution
(b) Use cylindrical shells to set up an integral for the volume of the top half of the pictured donut.
You should describe the region being rotated and describe what integration technique you would
use to find the volume.
3. (optional extra practice) You don’t need to turn these in.
ˆ Find the volume of the solid S whose base is
bounded by x = y+1 and x = y2−1, and the
cross sections ⊥ to the y-axis are rectangles
of height 4.
ˆ Set up integral(s) for the volume of the solid
obtained by rotating the region bounded by
y = x4, y = sin(πx/2) about x = −1.
ˆ Describe a solid of revolution whose
volume is given by the integral
∫ 5
1
π
(
(8− f(x))2 − (8− (−f(x)))2) dx.
ˆ Compare the area of the region R bound by
y = 1/x, and the x-axis on [1,∞) with the
volume of the solid obtained by rotating R
around the x-axis.
ˆ Use cylindrical shells to compute the volume
of a sphere.