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Solids of revolution aren't fruits thrown at tyrannical rulers in protest. A solid of revolution is a 3D object built by rotating an area around a predetermined center line called the axis of rotation.
We mentioned before that one way to think of this is as a bundt cake. If we aren't happy with a thin slice, we can choose a thicker piece by making an initial slice and rotating the knife around the center to cut out a better sized portion. If we were to rotate all the way around once without cutting, we would form the entire cake, which is a full revolution of the solid cake.
Understanding how to make solids of revolution can be difficult to picture, so drawing them is often very useful. Just like how football players practice plays before a big game, we're going to get plenty of practice drawing solids of revolution before we build integrals with them.
Draw the solid obtained by rotating the region bounded by y = x, y = 1, and the y-axis around the y-axis.
As always, we'll start by drawing the area we are going to make the volume with:
We can label the axis of rotation and draw a mirror copy of the region on the other side of the axis of rotation. Drawing the mirror copy helps us picture the axis of rotation more clearly. We know the y-axis is the axis of rotation because the problem said to rotate around the y-axis.
Next, we draw curved lines, solid in front and dotted in back, to make it look like the region went all the way around the y-axis:
This will usually be good enough to give you an idea of what the solid looks like. In this case we ended up with a cone, but different regions and different axes of rotation will produce radically different solids.
If you've ever been confused by whether it's spelled disk or disc, you're not alone. "Disk" is from the Greek diskos and "disc" is from the Latin discus, so go with whichever language you like better. The Greek or Latin to English lesson being over, let's toss around a few disks and see what volumes we get.
We're going to begin with a solid of revolution. We can slice it perpendicularly to the axis of rotation to get slices that look like disks or washers. If there's no hole in the middle of the slice, we call the slice a disk because it resembles a frisbee.
If there's a hole in the middle of the slice, we call the slice a washer because it looks like a washer used with a nut and bolt.
Here's a video that shows a solid of revolution with a washer-shaped slice taken out.
We make a disk by beginning with a region that touches the axis of rotation then slicing along the whole length of the solid. There won't be any holes in the middle of the solid, so all the slices will look like frisbees. Heads up!
To find the volume of the solid we first, we find the volume of a single disk. Then we figure out the appropriate limits of integration and write down an integral.
Because a disk is circular, we can find the volume of the disk by finding the area of its circular side and multiplying by its thickness. It's a thin disk, so it shouldn't hurt too much if we miss and it beans us in the face.
Let R be the region bounded by the curve y = x2, the x-axis, and the line x = 4. Write an integral expression for the volume of the solid obtained by rotating R around
(a) the x-axis
(b) the line x = 4
The region R looks like
(a) When we rotate R around the x-axis we get a solid that looks like a bell:
R lies along the x-axis for the entire length of the region. If we take slices perpendicular to the axis of rotation, each slice is a disk. The radius of the disk at position x is
y = x2.
This means the volume of that disk is
πr2 Δx= π(x2)2 Δ x = π x4 Δ x.
The variable x goes from 0 to 4 in the region R, which means we have disks from x = 0 to x = 4. We integrate from 0 to 4 to find the volume of the solid:
(b) Rotating R around the line x = 4 produces this solid:
Slices perpendicular to the line x = 4 are now horizontal. Since R lies along the line x = 4, each slice will be a disk. By looking at the slice at height y, we see from the line x = 0 to the edge of the slice is a distance of . (We just solved y = x2 for y.) From the edge of the slice to the center of the slice is a distance of r. Since the distance from x = 0 to the center of the slice is 4, we get
so the radius of the slice at height y is
Now the volume of the slice is
The variable y ranges from 0 to 16 in the region R, so we have disks from y = 0 to y = 16. This means the volume of the entire solid is
Be Careful: square the radii separately and then subtract. You want
π[(router)2 – (rinner)2],
π(router – rinner)2.
To figure out what the outer and inner radii are, we can draw pictures if it helps. We don't need any complicated formulas.
No, we aren't talking about Sally's seashells by the seashore. We are talking about another method for finding volumes of weird 3-D things by chopping them up into funny-shaped pieces, though. It's called the cylinder method or shell method. Instead of chopping the solid into disks or washers perpendicular to the axis of rotation, we carve it into cylindrical shells that surround the axis of rotation.
Let's get some visual help. We can take the region surrounded by , the line x = 1, and the x-axis .
In the above picture, we rotated that region around the y-axis, and then removed a cylinder-shaped shell from around the axis of rotation. Now we can label the radius and height of the cylinder to get
This shell is like a piece of paper that's rolled together so its edges just barely touch. It's called a shell because there's nothing inside it. Like a toilet paper tube, there's a little bit of stuff around the edge and a lot of empty space in the middle. Ack! We ran out of toilet paper.
We can find the volume of the solid by adding up the volumes of the infinitely many shells it's made of. This is similar to when we found the area of circle by chopping it into rings.
We have to answer a simple question first: what is the volume of a shell? If a shell is like a toilet paper tube, then we can unroll the tube. The height of the shell is the height of the unrolled cardboard. The circumference of the shell is the width of the unrolled paper. This means the area of one side of the paper is
The tube is very thin, with a thickness Δ r, where r is the variable describing the position of the shell. The volume of the piece of tubular shell, is going to be
(2πrh) Δ r.
To find the volume of the solid, we integrate over whatever values are reasonable for the variable of integration.
Let R be the region surrounded by , the line x = 1, and the x-axis. Use the shell method to find the volume of the solid obtained by rotating R around the y-axis.
We already know what the region R and the solid look like:
We can use x as the variable of integration. The shell at position x has radius x and height .
The volume of this shell is
We have shells from x = 0 to x = 1, the limits of integration. The volume of the solid is
When the axis of rotation isn't one of the coordinate axes, we have to do a little more work to figure out the radius of the shells. As with the disks and washers, we want to draw pictures to do this. It's very important.
If it looks like a duck, walks like a duck, and quacks like a duck, it must be a duck, right? Not in the case of shells and washers.
Even though we can use either method to do the same thing, we all know that metal washers and toilet paper tubes are nothing alike. One we are happy to use, while the other we are begrudged to see in what becomes a moment of desperation. The integration methods have some differences as well.
1) When using the washer method, we make cuts perpendicular to the axis of rotation. When using the shell method, we make our cuts parallel to the axis of rotation. Either way, the axis of rotation goes through the middle of the round thing. Got it?
2) When using the washer method, if we rotate around a horizontal axis we use y as the variable of integration. When using the shell method, if we rotate around a horizontal axis we use x as the variable of integration.
3) The two methods are based on different formulae. The washer method uses the formula for volume of a washer
π[(router)2 – (rinner)2] Δ x (or Δ y)
The shell method uses the formula for volume of a shell
2πrh Δ x (or Δ y)
In particular, the washer method involves squaring radii and the shell method doesn't.
Of course, we now have to ask which method is better. Would we rather use a washer or a toilet paper tube?
Most of the time we can use whichever method we like better. If a problem specifies a method, of course we should use that method (although if it's not the one we like and we have extra time, we could make sure both methods give us the same answer).
Every once in a while, there are some solids for which one method will be easier than the other.