Discs and Washers
Volumes of Revolution
Suppose you wanted to make a clay vase. It is made by
shaping the clay into a curve and spinning it along an axis. If we want to
determine how much water it will hold, we can consider the cross sections that
are perpendicular to the axis of rotation, and add up all the volumes of the
small cross sections. We have the following definition:
Find the volume of the solid that is produced when the region
bounded by the curve
Since we are revolving around the x-axis, we have that the cross section is in the shape of a disk with radius equal to the y-coordinate of the point.
Find the volume of the solid formed be revolving the region
between the curves
We draw the picture and revolve a cross section about the x-axis and come up with a washer.
The area of the Washer is equal to the area of the outer disk
minus the area of the inner disk.
A = p(2 - [x2]2) = p[x - x4]
In general, we have