How to Calculate Volumes Using Single Integrals

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  • 0:06 Volume of a Cone
  • 1:34 Area of a Slice
  • 6:40 Volume of a Pyramid
  • 8:16 Lesson Summary
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Lesson Transcript
Instructor: Kelly Sjol
Ever wonder where the equation for the volume of a cone comes from? Or the equation for the volume of a sphere? In this lesson, learn how to use a slicing technique to find the volume of a region by solving a single integral.

Finding the Volume of a Cone

The area of each slice depends on where you sliced through the cone
Cone Slices

Finding the volume of a can is pretty straightforward. You find the area of the base and you multiply it times the height. Finding the volume of a cone or a square cone is, perhaps, not always so obvious. Sure, we have a formula for it. But where did this come from?

We're going to learn about finding the volume using a slicing technique and integration. Imagine you've got this cone that you've tipped on its side. Along the height of the cone is the x-axis. It spears the cone in the middle.

If you could take a sheet of paper or a really sharp knife and slice down that cone, you would see a cross section, say, at the top of the cone, or for a smaller value of x somewhere in the middle of the cone or near the bottom of the cone. These cross sections would change. They'd be large at the top, smaller in the middle and even smaller at the bottom.

Let's say for this particular cone, though, that you were always looking at a circle in your cross section. The radius of that circle is going to depend on where you sliced it. In particular, for this exact cone, the radius is set at x. So, the radius of the cone is the height of the cone at that slice. And the angle between your cone and either the x or y-axis is going to be 45 degrees.

Finding the Area of a Slice

The volume of the cone is estimated by adding together the volumes of all the slices
Cone Volume

If I want to find the area of a particular slice, I know the radius is going to be r and r is going to be equal to x. The area of this slice, the area inside of the cone for any slice, is equal to pi times r^2 or pi times x^2. So, the area of every slice is going to depend on where exactly I sliced through the cone.

Okay, so my area's going to change. Near the top of the cone, it's going to be much larger. Near the bottom of the cone, the area that I find inside of my slice is going to be much smaller. How can I use this to find the volume?

What if I take the area of one of the slices and I multiply the area times the thickness of that slice? So, now I've got a little, teeny, tiny volume of a little, teeny, tiny slice. This is like if you cut a piece of salami up. You've got the big, gigantic roll of salami and you take a whole bunch of slices of it. If you want to find the volume of the whole salami, you just need the volume of each one of those slices and you add it all up. That's all we're doing here, except our salami is going from 1 inch in diameter to 5 inches in diameter.

Okay, so we're going to estimate the volume of our cone as the sum of all of the little volumes, all of the slice volumes. Each one of those slice volumes is going to be the cross-sectional area times the thickness of the slice. Here, I can write that as A as a function of x, because I know that my area changes as I go from near the bottom of the cone to near the top of the cone, times delta x. We're going to say that delta x is the thickness of my slice. That's the change in x between the top of my slice and the bottom, or between my two knife cuts.

I'm going to sum this up over all of these volumes. That is, for every single slice, from the first slice through the nth slice, if I take n slices.

I'm going to get a much better estimate of my volume if the slices are really small - if I find the volume of very, very small slices. This is opposed to one big slice. If I estimated the area of my entire cone, let's say, based on how thick it is at the top, I'm going to way overestimate. If I estimate the volume based on a slice at the bottom of the cone, I'm going to way underestimate.

Determining the volume by slicing is really using a Riemann sum
Volume Riemann Sum

The way to get around this is to take a lot of little, tiny, very thin slices. I'm going to take the limit, as my slice thickness goes to 0 over all of my slices, of the volume of each slice, that is A as a function of x, the cross-sectional area's function of x, times the thickness of the slice, delta x.

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