Surface integral example part 3: The home stretch
- [Instructor] So now that we have been able to express our d sigma, I think we're pretty close to evaluating the integral itself, and one thing I do wanna point out that might have been nagging you from the end of the last video, at the end of the last video, I took the principal root of cosine squared of t, and I simplified that to just being cosine of t. And you might have said, "Wait, wait, wait, wait, "What if cosine of t evaluated to a negative number? "If I then square it, it would be positive, "and then if I took the principal root of that, "I would then get the positive version of it. "I would essentially get the absolute value "of the cosine of t." And the reason why we were able to do this in particular in this video or in this problem is because we saw t takes on values between negative pi over two and positive pi over two. And so cosine of anything that's either in the first or the fourth quadrant, so this is t right over here, the cosine will always be positive for our purposes, for the sake of this surface integral. Cosine of t is always going to be positive, and so in this case, we don't have to write absolute value of cosine of t. We can just write cosine of t, and so hopefully that makes you satisfied. That was just based on how we parameterized the t. Now, that out of the way, let's actually evaluate the integral. Our original integral, the original integral, just to remind us, was the double, or I should say the surface integral, of x squared, d sigma. We already know what d sigma is. Now we just have to write x squared in terms of the parameters. Well, we know the parameterization of x. The x, in terms of the parameters right over here is cosine. This is our parameterization: x is going to be equal to cosine t, cosine of s. Let me write that down: x of s and t is equal to, I already forgot, I have a horrible memory, is equal to, we have to go back to the original parameterization, not these partial derivatives. Cosine t, cosine s. Cosine t, cosine s. Cosine t, and then cosine s, and we're taking the integral of that squared. So let's think about this a little bit. So let's just do this part right over here. If we square x, we're going to get cosine squared t, cosine squared s. Cosine squared s, that's the x squared part right over there, and then you have the d sigma, which is this stuff, which is times cosine, lemme do that same green, don't wanna confuse you with different shades of colors, times cosine of t, ds, dt. And now that we have this in terms of the parameters, the differentials of parameters, this essentially becomes a double integral with respect to these two parameters, and so, and the good thing is that the boundary is pretty straightforward with respect to s and t: s takes on all values, s takes on all values between zero and two pi, t takes on all values between negative two pi, and, sorry, negative pi over two and positive pi over two. So first, the way I wrote over here, we're gonna integrate with respect to s first, s goes between zero and two pi, and then t, lemme write and make clear, this is s, and then t will go between negative pi over two and positive pi over two. And so let's see if we can simplify this a little bit. This is equal to the double integral over that same region, over that same area, I guess we could call it, over that same area of, well now we have this cosine squared of t and then we have another cosine of t right over there, so lemme just right it this way, as cosine to the third of t times cosine squared... Cosine squared of s, and then ds, lemme color code it a little bit, ds, and so this is the integral for the ds part, and then dt. And this is when we integrate with respect to s, notice these two, the t parts and the s part, they're just multiplied by each other, so when we're taking the integral with respect to s, this cosine cubed of t really is just a constant, we can factor it out, and it could look something like this, so let me rewrite it, this could be the integral from, t goes from, I'll rewrite the boundaries, negative pi over two, to positive pi over two, cosine cubed of t, I just factor that out, and then I'll write the s part, times the integral, s is going to go between zero and two pi, and I'll write this in blue, cosine squared of s, ds, and then you have dt out there, you have dt, I'm gonna do the dt in green, gimme that same green, dt. And now, this outer sum we can view it, you essentially view it as the product, well, of all of this business right over here, this thing has no t's involved in it whatsoever, so we can rewrite this, and I'll write all the stuff involving the t's as green. So we can rewrite this as pi over two, from negative pi over two, to pi over two, cosine cubed of t dt times the integral, and I'm really just rearranging things, I guess you could kind of view this as the associative property, or I guess the commutative property. Well those things always confuse me, times the integral of zero to two pi of cosine squared of s, ds, and you didn't have to do it this way, you could've just evaluated it while it was kind of mixed like this, but this'll help us kind of work through the, the trigonometry a little bit easier. Now to solve these two integrals, we just have to resort to our trigonometry. Cosine squared of s, we can rewrite that as 1/2 plus one, actually let me do that in that same blue color so we don't get confused. That is the same thing as 1/2 plus 1/2 cosine of two s, and cosine cubed t, well that's the same thing, let's see, we can factor out a cosine of t, so let me rewrite, ah let's just do it, well let me just do it, both at the same time, just get all the trigonometry out of way. This right over here can be rewritten as cosine of t times cosine squared of t, and the intuition here is, if we can get a product of a sine doing something with a cosine, because cosine is sine's derivative, that's kind of, y'know, u-substitution, you see a function and its derivative, you can just kind of treat it as a variable, so that's what we're trying to get to right over here. So cosine squared of t can be rewritten as one minus sine squared of t, so this is cosine of t times one minus sine squared of t. And so we can rewrite this as cosine of t minus cosine of t, sine squared of t, and you might say "Wait, "this looked a lot simpler than this down here." That is true, it looks simpler, but it's easier to take the antiderivative of this, easier to take the antiderivative of cosine of t, and even over here, you have derivative of sine of t, which is cosine of t, and so essentially you can do u-substitution, which you probably can do in your head now. So let's evaluate each of these integrals. So this one, let me rewrite them just so we don't get too confused, so we have the integral from negative pi over two, to pi over two, of cosine of t minus cosine of t, sine squared of t, dt, times the integral from zero to two pi of 1/2 plus 1/2 cosine of 2s, d s. Now we are ready to take some antiderivatives, the antiderivative of this right over here is going to be, the antiderivative of cosine t, well that's just sine t, and then right over here, the derivative of sine t is cosine of t. So we can just essentially, if you wanna do u-substitution, you would say, u is equal to sine of t, d u is equal to cosine of t, dt, and you do all of that, but the, what we probably cannot do in your head is, okay, I have the sine t's derivative there, so I can treat sine t just like I would treat a t, or I would treat an x. So this is going to be, you still have this negative sign, minus sine to the third of t over three, if this was just a t squared, the antiderivative would be t to the third over three, but now since we have a derivative, we can kind of treat it the same way, which is essentially doing u-substitution in our head. So that's that, and we're going to evaluate it from negative pi over two, to pi over two. And so this is equal to, if you evaluate it at pi over two, sine of pi over two is one. So it's one minus 1/3, so that's just 2/3, actually lemme not write it that way, I don't wanna confuse people, and then minus sine of negative pi over two, well that's going to be negative one minus, sine of negative pi over two is negative one to the third power is negative one, so this is negative 1/3. And so this is going to be equal to, this is 2/3, and this is negative one plus 1/3, which is negative 2/3, but then you have a negative out front, so this is plus 2/3 again. So this part at least evaluates to 4/3. This part, all this, is really the home stretch, that all evaluates to 4/3. Now this part right over here, antiderivative of 1/2 is just 1/2 t, antiderivative of cosine of 2s, well, ideally you would have a two out front here, out front, lemme write and make this clear, so if I were to take the antiderivative of cosine of 2s, ideally you would want a two out here, so you have the derivative of the 2s, so you could put a two out front, but then you would have to put a 1/2 out front so that you're not changing the value of it, and of course you would have a ds right over here. I'm just taking a general antiderivative, but once you have it like this, then this just like taking the antiderivative of cosine of s. This becomes, antiderivative of cosine is sine. So this will become sine of s. So this right over here is just sine of s, and then you have the 1/2 out front, times 1/2, but then of course, and then you would have plus a constant if you were taking an indefinite integral, but we're taking a definite one, so you don't have to worry about the constant, so just the antiderivative of cosine of 2s, just the antiderivative of cosine of 2s is 1/2 sine of s. And so you have this constant out front, 1/2 times 1/2 is 1/4. So it's going to be plus 1/4 sine of 2s. That's the antiderivative, and now we're going to evaluate it from zero to two pi, and in either situation, this thing's going to evaluate to zero. Sine of zero is zero, sine of four pi is zero, and so you're gonna have 1/2 times two pi, which is just pi, plus zero, 'cause sine of four pi is zero, minus 1/2 times zero, zero; 1/4 times sine of zero, zero, so you're essentially just gonna end up with pi, so this whole thing right over here evaluates to pi. And so we're done! You take the product of these two things, 4/3 times pi, our entire surface integral evaluates to: 4/3 pi. So this is equal to 4/3 pi, which is neat! If you have a sphere of radius one, its surface area, or actually no, I shouldn't even go that, because, let me be very careful. I shouldn't make that statement, because this wasn't just with respect to one. But, we have at least evaluated the surface integral, and we deserve, I think, a bit of a rest now.