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:

where A(x) is the area the cross section at a point x.

Disks

Example

Find the volume of the solid that is produced when the region bounded by the curve

y = x2     y = 0    and     x = 2

is revolved around the x-axis.

Solution

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.

Hence

A(x) = pr2 = p[x2]2

We have

 Volume by Disks If the region below y = f(x), above the x-axis, and between the lines x  =  a and x  = b is revolved around the x-axis, then the volume of the resulting solid is given by

Example:  Washers

Find the volume of the solid formed be revolving the region between the curves

y = x2   and   y =

Solution

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(R2 - r2)

We have that R is the y-coordinate of the top curve (y = ) and r is the y-coordinate of the bottom curve (y = x2).  We have

A  =  p(2 - [x2]2)  =  p[x - x4]

Hence

In general, we have

 Volume by Washers If the region below y = f(x), above the y  =  g(x), and between the lines x  =  a and x  = b is revolved around the x-axis, then the volume of the resulting solid is given by

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