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Welcome to Calculus.

Â I'm Professor Ghrist.

Â We're about to begin lecture 32 on simple volumes.

Â In this lesson we'll progress from computing areas in 2D

Â to computing volumes in 3D.

Â Following the pattern that we use with areas, we'll begin with some of

Â the classical shapes and consider their volume formulae.

Â We'll look at round balls, cones and pyramids.

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The simplest example would be that of a rectangular

Â prism of length l, width w and height h.

Â In this case, computing the volume element is simple if

Â we slice along one of the three principal directions.

Â One volume element might be w times hdx, integrated from 0 to l.

Â Another l times hdy, integrated from 0 to w.

Â Or finally l times wdz, integrated from 0 to h.

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If instead we use an angular wedge, then we integrate with respect to theta,

Â the volume element, 1/2 r squared h d theta.

Â Here theta goes from 0 to 2 pi.

Â Finally, if we use an annular area element,

Â then, associating a t variable to that.

Â We consider as a volume element this cylindrical shell,

Â whose volume is 2 pi t times h dt.

Â Here we integrate from 0 to r.

Â Now moving up in complexity a little bit, consider a round ball of radius r.

Â 3:20

In this case, we're going to slice along a lateral plane.

Â This leads to a disc shaped volume element.

Â But as we slide the plane back and forth over this solid ball,

Â we see that the radius of the disc changes.

Â It's going to take a little bit of work to figure out this volume element.

Â In this case since it is always a disc, the volume is the area of that disc,

Â Pi times whatever its radius is squared, times the thickness.

Â Let's call it dx.

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Now, at a particular value of x, what is the associated radius?

Â It will help to build a right triangle in a cross-sectional view

Â whose base is of length x, whose radius is of course r.

Â What is the height?

Â In this case, it is square root of r squared- x squared.

Â So, to compute the volume, we integrate the volume element.

Â That is, we integrate pi times quantity r squared- x squared dx.

Â As x goes from negative r to r.

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In contrast with computing the area of a round disc of radius r,

Â computing the volume of a round ball of radius r is relatively simple.

Â This integral is very straightforward,

Â yielding pi times (r squared x -1/3 x cubed).

Â Evaluating that from negative r to r leads, with just a little bit of algebra,

Â to the familiar formula 4/3 pi r cubed.

Â There's more than one way to derive this volume formula.

Â Let's look at a different method.

Â Consider slicing up the ball into cylindrical shells.

Â As a volume element, let's choose a radial variable.

Â Let's call it t.

Â And then consider a cylinder.

Â That sweeps out the volume of the ball parallel to some, let's say vertical axis.

Â In this case, what can we say about the volume element?

Â Well, I claim it is 2 pi th dt.

Â 6:21

Again, building the associated right triangle is going to help us out.

Â This triangle in profile, is going to have hypotenuse r, the radius of the ball.

Â The base is of length t, the radius of our cylinder.

Â The height is given by Pythagoras as the square root of r squared- t squared.

Â But that's the height of the triangle.

Â The height of a cylinder is this doubled.

Â Hence, our volume element is 4 pi t.

Â Square root of r squared- t squared dt.

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And to compute the volume of the ball in this alternate fashion.

Â We integrate this volume element as t goes from 0 to r.

Â Now this is not so difficult of an integral.

Â It's not quite as easy as the last one, but a simple substitution.

Â Letting u be r squared- t squared,

Â we see that du is -2tdt.

Â And that we can rewrite this as the integral of -2 pi square root of u du.

Â As u goes from r squared to 0.

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We're going to need to choose a variable to denote the radius of this shell.

Â Let's call that rho.

Â That's a letter that you'll be seeing again in multi-variable calculus.

Â In this case, I claim that the volume element

Â is really the surface area of this shell,

Â 4 pi rho squared, times the thickness, d rho.

Â Now, I'm not going to justify that computation,

Â other than to demonstrate that when we integrate it as rho goes from 0 to r,

Â we obtain, most simply, 4/3 pi rho cubed.

Â As rho goes from 0 to r, we have, very easily, the formula for the volume.

Â 10:07

And then, consider slicing along disks that are parallel to the x-axis.

Â All of these slices are going to be discs of some radius.

Â And so the volume element is going to be pi times x

Â squared dy, where x is the radius of this disc.

Â But we want to integrate with respect to y since our thickness,

Â our differential element, is dy.

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So, writing things as we have done so,

Â with the apex of the origin of the plane.

Â We see that the equation for

Â the line that determines the profile of the cone is y = h/r x.

Â Solving for x as r over h times y, we substitute into the formula for

Â the volume element to obtain pi times quantity

Â (r/h y) squared dy.

Â And now to compute the volume, we integrate this

Â volume element as y goes from 0, the bottom to h, the top.

Â Pulling out the associated constants gives pi r squared/h squared.

Â What's left is the integral of y squared dy, which is, of course, y cubed over 3.

Â Evaluating from 0 to h yields the perhaps

Â familiar formula 1/3 pi r squared h.

Â Now what happens if,

Â instead of a cone with a circular base, we consider something else.

Â Let's say a square, side length s.

Â Proceeding as before by turning the cone upside down and

Â slicing along lateral planes.

Â We see that the volume element is a thickened square of some side length.

Â What is that side length?

Â If we solve for the line that constrains y in terms of x and solve for

Â x, then we obtain a volume element that is 4 x squared dy.

Â Namely, 4 times quantity (s/2h y) squared dy.

Â Integrating that to obtain the volume leads to the integral

Â of s squared over h squared times y squared dy.

Â As y goes from zero to h.

Â Pulling out the constants we are left with the integral of y squared dy,

Â yielding y cubed/3.

Â Evaluating from zero to h and multiplying by those constants out in front

Â leads to 1/3 s squared h.

Â You may sense a pattern here.

Â In both cases, it was one-third the area of the base times the height.

Â Well, it turns out that this is true in general for

Â any cone with a base of area B, and height h.

Â No matter what its shape, we can, by turning it upside down and

Â slicing, obtain copies of that base.

Â Copies that are rescaled by some factor.

Â So that the volume element is B, the area of the base,

Â times some factor, times the thickness dy.

Â The question is, what is that scaling factor as a function of y?

Â Well, let's say that you go halfway down.

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At y equals h over 2, what is happening to the area of the slice?

Â The area has not decreased by one-half.

Â Rather, it is multiplied by a factor of one-fourth, that is,

Â one-half squared because we're rescaling both directions.

Â Therefore, the appropriate volume element is B (y/h) squared dy.

Â And when we integrate that, we see that the constants B and

Â h squared come out in front and what's left is the integral y squared dy.

Â