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This worksheet explores methods for computing volumes, specifically the disk and washer methods, within the context of calculus ii. It includes exercises on drawing three-dimensional objects, finding cross-sectional areas, and determining volumes using integration. A structured approach to understanding volume calculation through practical examples and geometric interpretations, suitable for students learning integral calculus.
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This week we explore two methods of computing volumes: The Disk Method and The Washer Method.
In quick summary, area is a two dimensional measurement of how much space an object takes up.
Volume is a three dimensional measurement of how much space an object takes up.
a) A cylinder
b) A sphere
c) A pyramid with a square base
A cross section of a three dimensional object gives you an area at a particular section. The completed object can be thought of as a collection of cross sections "glued" together.
a) A cylinder
b) A sphere
c) A pyramid with a square base
In the case that your cross-section is a perfect circle (which happens fre- quently when you revolve a function about an axis), then you can backpedal to geometry to make this even simpler. This is known as the disk method.
With the disk method, the volume is computed just as it was before, but with the additional information you know for each cross-sectional area.
V (x) =
Z (^) b
a
A(x)dx =
Z (^) b
a
dx
Building on the disk method will give us another interesting way to com- pute a volume called the washer method. Suppose that the disk has been hollowed out to resemble a washer.
Figure 1: A washer
Because you want to remove a portion from the disk you've already gath- ered, we will again use the mysterious art of subtraction.
We can again compute the volume of the object just as we did before with our new information
V (x) =
Z (^) b
a
A(x)dx =
Z (^) b
a
dx
