Welcome to Calculus. I'm Professor Ghrist. We're about to begin lecture
32 on simple volumes. In this lesson we'll progress
from computing areas in 2D to computing volumes in 3D. Following the pattern that we use
with areas, we'll begin with some of the classical shapes and
consider their volume formulae. We'll look at round balls,
cones and pyramids. When computing volumes, there's really
only one formula you need to know. And that is that the volume is
the integral of the volume element, and all of the difficulty lies in
finding the correct volume element. The simplest example would
be that of a rectangular prism of length l, width w and height h. In this case,
computing the volume element is simple if we slice along one of the three
principal directions. One volume element might be w times hdx,
integrated from 0 to l. Another l times hdy,
integrated from 0 to w. Or finally l times wdz,
integrated from 0 to h. All three integrals return the volume,
length times width times height. For a less trivial example,
consider a cylinder with base a disk of radius r and height h. There are several ways
to compute the volume. There's pi r squared h. One method involves taking
the associated area element for the base and multiplying by
the height of the cylinder, h. As we saw in a previous lesson on area, there are several
different ways to do this. We could use a lateral
slice of thickness of dx. And obtain as a volume
element 2h square root of r squared- x squared dx,
integrating from -r to r. If instead we use an angular wedge,
then we integrate with respect to theta, the volume element,
1/2 r squared h d theta. Here theta goes from 0 to 2 pi. Finally, if we use
an annular area element, then, associating a t variable to that. We consider as a volume element
this cylindrical shell, whose volume is 2 pi t times h dt. Here we integrate from 0 to r. Now moving up in complexity a little bit,
consider a round ball of radius r. In this case, we're going to
slice along a lateral plane. This leads to a disc
shaped volume element. But as we slide the plane back and
forth over this solid ball, we see that the radius
of the disc changes. It's going to take a little bit of work
to figure out this volume element. In this case since it is always a disc,
the volume is the area of that disc, Pi times whatever its radius is squared,
times the thickness. Let's call it dx. Now, at a particular value of x,
what is the associated radius? It will help to build a right
triangle in a cross-sectional view whose base is of length x,
whose radius is of course r. What is the height? In this case,
it is square root of r squared- x squared. So, to compute the volume,
we integrate the volume element. That is, we integrate pi times
quantity r squared- x squared dx. As x goes from negative r to r. In contrast with computing the area
of a round disc of radius r, computing the volume of a round ball
of radius r is relatively simple. This integral is very straightforward, yielding pi times (r
squared x -1/3 x cubed). Evaluating that from negative r to r
leads, with just a little bit of algebra, to the familiar formula 4/3 pi r cubed. There's more than one way to
derive this volume formula. Let's look at a different method. Consider slicing up the ball
into cylindrical shells. As a volume element,
let's choose a radial variable. Let's call it t. And then consider a cylinder. That sweeps out the volume of the ball
parallel to some, let's say vertical axis. In this case,
what can we say about the volume element? Well, I claim it is 2 pi th dt. That is the area of the annular
strip of thickness dt and radius t times the associated
height h of the cylinder. Well, that's great, except what is h? It depends on t. Again, building the associated right
triangle is going to help us out. This triangle in profile, is going to have
hypotenuse r, the radius of the ball. The base is of length t,
the radius of our cylinder. The height is given by Pythagoras as
the square root of r squared- t squared. But that's the height of the triangle. The height of a cylinder is this doubled. Hence, our volume element is 4 pi t. Square root of r squared- t squared dt. And to compute the volume of
the ball in this alternate fashion. We integrate this volume
element as t goes from 0 to r. Now this is not so
difficult of an integral. It's not quite as easy as the last one,
but a simple substitution. Letting u be r squared- t squared, we see that du is -2tdt. And that we can rewrite this as
the integral of -2 pi square root of u du. As u goes from r squared to 0. That integrates, simply, to negative
2 pi times u to the 3/2 over 3/2. Evaluating from r squared to 0 and
simplifying the algebra. Leads to 4/3 pi r cubed. A last method for decomposing the ball and obtaining the volume would be to
use a spherical volume element. This is a round shell. Concentric with the ball. We're going to need to choose a variable
to denote the radius of this shell. Let's call that rho. That's a letter that you'll be seeing
again in multi-variable calculus. In this case,
I claim that the volume element is really the surface area of this shell, 4 pi rho squared,
times the thickness, d rho. Now, I'm not going to
justify that computation, other than to demonstrate that when we
integrate it as rho goes from 0 to r, we obtain, most simply, 4/3 pi rho cubed. As rho goes from 0 to r, we have,
very easily, the formula for the volume. Using this spherical shell,
seemed to be the simplest integral. But determining what the volume element is in terms of the surface area of
that shell, was maybe not so simple. The moral of this story is that the proper volume element can make
the integration very, very simple. Let's switch to a different example,
and consider cones. You may recall the formula for
the volume of a cone with a circular base of radius r, and height h. Let's turn a cone upside down, putting the point at the origin
of the x y plane in profile. And then, consider slicing along disks
that are parallel to the x-axis. All of these slices are going
to be discs of some radius. And so the volume element
is going to be pi times x squared dy,
where x is the radius of this disc. But we want to integrate with
respect to y since our thickness, our differential element, is dy. So, writing things as we have done so, with the apex of the origin of the plane. We see that the equation for the line that determines the profile
of the cone is y = h/r x. Solving for x as r over h times y,
we substitute into the formula for the volume element to
obtain pi times quantity (r/h y) squared dy. And now to compute the volume,
we integrate this volume element as y goes from 0,
the bottom to h, the top. Pulling out the associated constants
gives pi r squared/h squared. What's left is the integral of y squared
dy, which is, of course, y cubed over 3. Evaluating from 0 to h yields the perhaps familiar formula 1/3 pi r squared h. Now what happens if, instead of a cone with a circular base,
we consider something else. Let's say a square, side length s. Proceeding as before by turning
the cone upside down and slicing along lateral planes. We see that the volume element is
a thickened square of some side length. What is that side length? If we solve for the line that constrains
y in terms of x and solve for x, then we obtain a volume
element that is 4 x squared dy. Namely, 4 times quantity
(s/2h y) squared dy. Integrating that to obtain
the volume leads to the integral of s squared over h squared
times y squared dy. As y goes from zero to h. Pulling out the constants we are left
with the integral of y squared dy, yielding y cubed/3. Evaluating from zero to h and multiplying
by those constants out in front leads to 1/3 s squared h. You may sense a pattern here. In both cases, it was one-third
the area of the base times the height. Well, it turns out that
this is true in general for any cone with a base of area B,
and height h. No matter what its shape, we can,
by turning it upside down and slicing, obtain copies of that base. Copies that are rescaled by some factor. So that the volume element is B,
the area of the base, times some factor, times the thickness dy. The question is, what is that
scaling factor as a function of y? Well, let's say that you go halfway down. At y equals h over 2, what is
happening to the area of the slice? The area has not decreased by one-half. Rather, it is multiplied by
a factor of one-fourth, that is, one-half squared because we're
rescaling both directions. Therefore, the appropriate volume
element is B (y/h) squared dy. And when we integrate that,
we see that the constants B and h squared come out in front and
what's left is the integral y squared dy. Evaluating that from 0 to h yields
1/3 the base times the height. And so we see that the real reason for
that 1/3 in the volume of a cone comes from integrating y squared. That's a nice insight to
this classical formula. And so we see that the classical
volume formulae still hide insights. Now we know why there's a one-third
in the volume of a cone and why there's a four-thirds
pi in the volume of a ball. In our next lesson, we're going to turn
from these simple volume formulae, and consider some more complex objects.