Solving Volume Problems

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The vertices of the base are joined by segments creating lateral faces shaped like triangles that meet at one point above the base of the pyramid. You may also see that point called the apex of a pyramid.

For example, the Pyramid of Cestius, in Rome, Italy, is an example of a pyramid with a rectangular base.

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.

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.There are many solids out there that cannot be generated as solids of revolution, or at least not easily and so we need to take a look at how to do some of these problems.Now, having said that these will not be solids of revolutions they will still be worked in pretty much the same manner.\right|_0^h = \pi h\] So, we get the expected formula.Also, recall we are using \(r\) to represent the radius of the cylinder.All of the examples in this section are going to be more general derivation of volume formulas for certain solids.As such we’ll be working with things like circles of radius \(r\) and we’ll not be giving a specific value of \(r\) and we’ll have heights of \(h\) instead of specific heights, .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. Due to the nature of the mathematics on this site it is best views in landscape mode.If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

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