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All other letters in the integral should be thought of as constants.If you have trouble doing that, just think about what you’d do if the \(r\) was a 2 or the \(h\) was a 3 for example.While \(r\) can clearly take different values it will never change once we start the problem.
You can see in the picture that the lateral faces are triangles, and that the edges of the lateral faces all meet at one point at the top, or vertex, of the pyramid.
In this lesson, you solved problems involving the volume of rectangular pyramids and triangular pyramids.
\right|_0^h = \pi h\] So, we get the expected formula.
Also, recall we are using \(r\) to represent the radius of the cylinder.
Let’s start with a simple example that we don’t actually need to do an integral that will illustrate how these problems work in general and will get us used to seeing multiple letters in integrals.
Now, as we mentioned before starting this example we really don’t need to use an integral to find this volume, but it is a good example to illustrate the method we’ll need to use for these types of problems.
What we need here is to get a formula for the cross-sectional area at any \(x\).
In this case the cross-sectional area is constant and will be a disk of radius \(r\).
In this section we’re going to take a look at some more volume problems.
However, the problems we’ll be looking at here will not be solids of revolution as we looked at in the previous two sections.