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
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
about the x-axis.
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]
Example: Revolving about the y-axis
Find the volume of the solid that is formed by revolving the
curve bounded by
A = p((y)2 - (y2)2) = p(y2 - y4)
Revolving About a Non-axis Line
Find the volume of the region formed by revolving the curve
A = p(2 + x3)2
This integral can be evaluated by FOILing out the binomial and then integrating each monomial. We get a value of approximately 133.
Try revolving the curve
A = p[(5 - )2 - 9]
This integral works out to be approximately 59.
Applications of Volume
Example: The Volume of the Khufu Pyramid
The base of the Khufu pyramid is a square with wide length 736 feet and the angle that the base makes with the ground is 50.8597 degrees. Find the volume of the Khufu pyramid.
50.8597 degrees = .88767 radians.
The height of the pyramid is
We have that the area of a cross section is s2 where s is the side length of the square.
Placing the y-axis through the top of the pyramid and the origin
at the middle of the base, we have that
We set up similar triangles:
x/736 = y/904.348
x = .8138y
= 361131 cubic feet.
Example: Volume of a Sphere
A sphere is formed by rotation the curve
Hint: Consider cross sections parallel to both axes of rotation. These cross sections are squares. Then show that the side length is