Computing Volumes: Cross-Sectional Method
Disc Method: Revolve the graph of a continuous non-negative function f on an interval [a,b]
over the x-axis.
V =
1. Find the volume of the solid obtained by revolving f(x) = over the x-axis for the
interval [0, ]
2. Find the volume of the solid obtained by revolving f(x) = over the x-axis for the
interval [0, 1]
Washer Method: Revolve the region between the functions f and g on an interval [a,b] over the
x-axis.
V =
(Assume f(x) > g(x) > 0)
3. Find the volume of the solid obtained by revolving the region between f(x) =
and g(x) = on the interval [1,3] about the x-axis. (Sketch the region as part of
your answer.)
4. Find the volume V of the solid generated by revolving the area between the graphs of
and (note: you need to determine the appropriate interval for
this problem). (Sketch the region as part of your answer.)
5. Suppose a washer for a faucet has the shape of the solid generated by revolving about
the x-axis the figure bounded by the y-axis, the lines x = 1 and y = ½, and the curve
Sketch the region. Compute the volume.