Welcome back. I don’t know what I was thinking.
The following video will tell you about Solid of Revolution (part 3). You can use youtube to mp3 tool to convert the video to mp3.Notice: download videos and mp3 must be licensed by the version owner and used only for study and research, and not for commercial use and dissemination.
Solid of Revolution (part 3)
Sometimes my brain malfunctions. But just going back to that problem we were doing, actually I think we should do it. I’m a little schizophrenic today. So let’s figure out the equation for the volume of a sphere.
So what’s the equation? It’s x squared plus y squared is equal to r squared. And let’s just write y as a function of x, just so we can do it the way we did that last problem. So you get y squared is equal to r squared minus x squared. y is equal to the square root of r squared minus x squared.
And let’s draw it. So if this is my y-axis, this is my x-axis, and the equation– draw it straight– that’s my x-axis, and then I actually have a circle tool, let me see if I can use it effectively– well, close enough. There you go. I think you get the point.
But anyway. This is just going to be the upper half of the circle. Actually, I should probably just undo that circle tool and try to draw it by hand. So y equals the square root of r squared minus x squared.
That’s just going to be the upper half of the circle. So it will be the positive x quadrant– and then actually, I should have drawn the whole hemisphere. Actually, let me do that, because I think it’ll make– edit, undo, let me clear all of this out.
Sorry for wasting your time, but I think it’ll be effective. OK. So let me redraw. So this, that’s the y-axis, that’s my x-axis, and then this– the square root is, since it’s a function, it can only have one value, so we assume it’s defined as the positive square root.
So if we were to graph that, it would look like this. Something like that, where this would be minus r and that’s r. So if we want to find the volume of a sphere with radius r, we just have to rotate this function around the x-axis.
This is the x-axis, that’s the y-axis. So let’s see what we can do. Let’s just visualize the disks again. So let me make a disk.
So let’s say that that’s the side of one of the disks again, and as we know, the depth of the disk is just going to be dx. That’s how wide that disk is, dx. And its radius at any point is f of x, and in this case, it’s y is equal to square root of r squared minus x squared. So what’s the surface area of each disk? What’s this? The surface area of each of the disks.
I hope you know what I’m saying. So area is equal to pi r squared, the radius at any point is equal to this, radius is equal to y which is equal to square root– and remember, this is not this r. This is the radius of this disk. I know it might be a little confusing.
So the area is going to equal pi times this squared. So if you square this quantity, you just get rid of the square root sign, right? So pi r squared minus x squared, and that’s the area, and so what’s the volume of that disk? Well just like we’ve done in every video up to this point, the volume of that disk is just that, so the volume of that disk is just this pi r squared minus x squared times dx. And so if we want to figure out the volume of all these disks, I have a disk here, a disk here, going around and around and around and around and around and they get smaller and smaller until we have a sphere. We just take the integral, the upper bound is positive r, the lower bound is minus r, and we take the integral of this expression.
pi– let me distribute it, because that’s going to make it easier– pi r squared, which is just a constant term, minus pi x squared, all of that dx. So what’s the antiderivative of that expression? The antiderivative within the parentheses. Well, this is just a constant term.
pi r squared, that’s just a number, because we’re just taking the integral with respect to x. So the antiderivative of pi r squared is just pi r squared x, the derivative of pi r squared x is just pi r squared, minus– and we did this in the last video. Actually, well now, it’s the antiderivative x squared, which is x to the third over 3, and the pi is just a constant, so pi x to the third over 3, and we’re going to evaluate that at r and minus r.
Let me erase some stuff, looks like I’m running out of space. Hopefully all of that you know by now. OK, back to the pen tool.
So let’s evaluate it at r. So this is pi r squared, and then for x, we’ll substitute the positive r times r minus pi x cubed, but now we have this r here, so r cubed over 3 minus pi r squared, and then we have a minus r here, because we’re evaluating the antiderivative at minus r, times minus r, minus pi minus r cubed. So what’s minus r cubed? It’s r cubed, but we’ll keep the minus sign. r cubed, and at that minus sign, let’s just make that– that’ll turn that into a plus– over 3.
Let’s see if we can clean this up a little bit. So that first term is pi r cubed, r squared times r, minus essentially 1/3pi r cubed. And then, what is this? This is pi r cubed, but then we have a minus sign up.
This is minus pi r cubed, and then we have a minus sign up here, so this becomes plus pi r cubed, and then minus– because we have a plus here and a minus out here, so distribute it– so minus 1/3pi r cubed. And let’s see, what do we have? We have essentially 1. If we just distribute out the pi r cubes, we have pi r cubed times 1 minus 1/3 plus 1 minus 1/3. Well that’s 2 minus 2/3, or another way, let’s see, is 2 minus 2/3– this is turning into a fractions problem– and what’s– well, that’s 6/3 minus 2/3, it equals 4/3.
So this is equal to 4/3. So the volume of the sphere is equal to 4/3 pi r cubed, which is the equation for the volume of a sphere. And actually, now that I realize, it did take me eight and a half minutes, so I am glad.
My first intuition is always correct, I am glad I did this in a separate video. But that should be pretty interesting to you. And it makes a lot of sense.
It’s going to be a cube of the radius, pi is involved. The 4/3 is interesting, just in terms of how it relates to everything else. Area is pi r squared, and then all of a sudden you get a 4/3 here, so it is something for you to think about.
Anyway, hopefully you found that fun. I’ll see you in the next video.
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