This document discusses calculating volumes of solids generated by revolving regions bounded by graphs of functions around axes. It contains three problems, the first asks to find the area of two regions R and S bounded by graphs in the first quadrant and revolving S around the x-axis to find its volume. The second problem asks to find the area and revolved volume of a region R bounded by graphs. The third asks to find the volume of a solid with the region R as its base and cross-sections that are squares.

1. Calculating Volumes
or the Calculus of sushi
Sushi Lunch by flickr user Harris Graber

2. Let R and S be the regions in the first
quadrant shown in the figure above. The
region R is bounded by the x-axis and the
graphs of y = 2 - x 3 and y = tan x . The
region S is bounded by the y-axis and the
graphs of y = 2 - x 3 and y = tan x.
(a) Find the area of R.
(b) Find the area of S.
(c) Find the volume of the solid generated when S is revolved
about the x-axis.

3. Let R be the region bounded by the the y-axis
and the graphs of the two functions shown in
the figure.
(a) Find the area of R.
(b) Find the volume of the solid
generated when R is revolved about the
x-axis.
(c) The region R is the base of a solid. For this solid each cross
section perpendicular to the x-axis is a square. Find the volume of
the solid.