In this video, we're going to focus on finding the volume using the shell method. So let's begin by drawing a picture. Now we're going to find the volume when rotated about the y-axis.
What I'd like to do is draw a rectangle. The limits of integration is going to be from A to B. The radius is the distance between the x-axis and the axis of rotation. By the way, you should always draw the rectangle parallel to the axis of rotation.
So in this case, you want it to be parallel to the y-axis. And then this is going to be the height of the shell, or the rectangle. So using the shell method, the equation is this. The volume is going to be equal to 2 pi. Integration.
from a to be and then it's the radius which is going to be in terms of X times the height in terms of X so if you want to find the volume others solid when rotated about the y-axis, the radius and the height have to be in terms of x. Now sometimes you may have two curves as opposed to one. Let's call the top curve f and the bottom g.
So we want to rotate about the y-axis. So the rectangle is going to be parallel to the y-axis. Now the radius is still the distance between the rectangle and the axis of rotation. And this is going to be the height of the shell.
The only difference in this situation is that the height, h of x, is basically equal to the top function, f of x, minus the bottom function, g of x. Everything else being the same. The limits of integration will still be a to b.
They will represent x values. Now sometimes, you may need to find a volume when rotated about the x-axis. So let's use the same type of picture, but we're going to rotate the region about the x-axis this time, instead of the y-axis.
So now we're going to draw the rectangle parallel to the axis of rotation, that is parallel to the x-axis. The radius is the distance between the rectangle and the axis of rotation. The height is parallel to the axis of rotation, so this is the height of the rectangle.
So the volume is going to be pi, well 2 pi, not just pi. And this time, the limits of integration is c to d. It's going to be y values.
So 2 pi integration from c to d. of the radius in terms of y times the height in terms of y, dy. So keep this in mind.
If you want to rotate the curve about the x-axis, the radius and the height has to be in terms of x for this to work. Now let's see if we have two functions, f and g. Let's say the function on the right is f and the one on the left is g.
We're going to rotate about the x-axis. So let's draw the rectangle parallel to the x-axis. This is going to be the radius still. And here we have the height. So the height is going to be equal to the function on the right, which is f of x, minus the function on the left, g of x.
And that's it. That's how you define h. So let's start with the first example.
Let's say that we have the curve y is equal to the square root of x, and the region is bounded by that curve, the line y equals 0, and x equals 4. And we're going to rotate it about the y-axis. So feel free to pause the video and try this problem. So let's begin by drawing a picture.
So this is the square root of x, and it's going to stop at x equals 4. So let's draw a rectangle, and we're going to rotate this about the y-axis. So this is the radius, and this is the height. So notice that the radius is the same as x. So thus we can say that r is equal to x.
x is the distance between the y-axis and the rectangle. Notice that the height h is the same as y. So we can say that h is equal to y.
Now, if we're rotating about the y-axis, or about any line parallel to the y-axis. The radius and the height has to be in terms of x. Now the radius is already in terms of x, which is good.
We just got to convert y into something in terms of x. Now we know that y is equal to the square root of x, so we can replace y with root x. So now we have h in terms of x.
Now let's use the formula. So the volume is equal to 2 pi integration from a to b, r of x times h of x dx. a is 0, b is 4. So we're going to integrate it from 0 to 4. The radius is x, the height is the square root of x. X is basically X to the first power. Square root of X is X to the one half.
Whenever you multiply it by common bases, you need to add the exponents. So 1 plus one half. That's basically 2 over 2 plus 1 over 2. That's 3 over 2. So therefore, we can rewrite the expression like this.
This is what we have so far. So now let's integrate it. The antiderivative of x to the 3 half, you need to add 1 to the exponent.
3 over 2 plus 1 is 5 over 2. And instead of dividing by 5 over 2, let's multiply by 2 over 5. Let's evaluate it from 0 to 4. So first, let's plug in 4. So we have 4 raised to the 5 over 2, multiplied by 2 over 5. And then minus 2 times 0, raised to the 5 over 2. over 2 over 5 which the whole thing is going to be 0. Now what is 4 raised to the 5 over 2? This is the same as 4 to the half raised to the fifth power. Whenever you raise one exponent to another you need to multiply the two exponents. 1 half times 5 is 5 over 2. Now 4 to the 1 half power is the same as the square root of 4 so that's 2 and 2 to the fifth power is 32. So we have 2 pi.
2 times 32 over 5 minus 0 and 2 times 32 is 64 and 2 times 64 is 128 so the final answer is 128 pi divided by 5 and so that's the solution Now let's work on another example. Let's say that y is equal to x minus x cubed and the region is bounded by that curve, the line y equals 0, and it's from x equals 0 to x equal 1. And we're going to rotate this region about the y-axis. So try this example. See if you can get the right answer, using what you know already. Now, what we can do is, let's find the x-intercepts for this graph.
Let's set y equal to 0. If we factor on an x, we're going to have 1 minus x squared left over. And we could factor 1 minus x squared using the difference of perfect squares method. It's 1 plus x and 1 minus x. The x-intercepts are 0, negative 1, and 1. It turns out that if you graph this function, and...
Focusing only on the right side of the y-axis. This graph is going to look something like this. Where this is 0 and this is 1. We're going to rotate it about the y-axis.
And so let's draw a rectangle that's parallel to the y-axis. So this is going to be the radius, and here we have the height. So just like before, the radius is the same as x, and the height we can see is y.
So, r equals x, and h is equal to y. But we need the radius and the height to be in terms of x. So, let's replace y with x minus x cubed. So, that's going to be the height.
So now we can write the integral. The volume is going to be 2 pi, integration from a to b, or 0 to 1, and then the radius times the height, dx. So what we're going to do at this point is distribute x. x times x is x squared.
x times x cubed is x to the fourth. So now let's integrate the function. The antiderivative of x squared is x cubed over 3 and the antiderivative of x to the fourth is x to the fifth over 5 integrated from 0 to 1 and let's not forget to multiply our answer by 2 pi.
Now let's begin by plugging in 1. 1 to the 3rd is just 1, so it's going to be 1 over 3. And 1 to the 5th is 1, so 1 over 5. And then if we plug in 0, 0 cubed is 0, 0 to the 5th is 0, so the whole thing is 0. Now we need to get common denominators. So let's multiply 1 over 3 by 5 over 5. And then let's multiply 1 over 5 by 3 over 3. So this will give us a common denominator of 15. So what we have now is 5 over 15 minus 3 over 15. 5 minus 3 is 2. So this is 2 over 15 times 2 pi. 2 times 2 pi is 4 pi. So the final answer to this problem, it's 4 pi over 15.