Solving Volume Problems

While \(r\) can clearly take different values it will never change once we start the problem.Cylinders do not change their radius in the middle of a problem and so as we move along the center of the cylinder ( the \(x\)-axis) \(r\) is a fixed number and won’t change.

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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.

We’ll start off with the sketch of the cylinder below.

We’ll center the cylinder on the \(x\)-axis and the cylinder will start at \(x = 0\) and end at \(x = h\) as shown.

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.Also, before we proceed with any examples we need to acknowledge that the integrals in this section might look a little tricky at first.There are going to be very few numbers in these problems.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.Note that we’re only choosing this particular set up to get an integral in terms of \(x\) and to make the limits nice to deal with.There are many other orientations that we could use.All the letters in the integrals are going to make the integrals look a little tricky, but all you have to remember is that the \(r\)’s and the \(h\)’s are just letters being used to represent a fixed quantity for the problem, it is a constant.So, when we integrate we only need to worry about the letter in the differential as that is the variable we’re actually integrating with respect to.For the prism shown at right, the base has an area of 12 square units.Since the height of the prism is 5 units, there are 5 layers required to build the full prism, so the volume of the prism is 12 × 5 = 60 cubic units.

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