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Find the volume of the solid obtained by rotating about the line $ x = 1 $ the region under the curve $ y = x \sqrt{1 - x^2} $ , $ 0 \le x \le 1 $.
$\frac{\pi(16-3 \pi)}{24} \approx 0.861$ units $^{3}$
Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 3
Trigonometric Substitution
Integration Techniques
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Let's find the volume of the solid obtained by rotating about the line X equals one. That's this green line right here. The vertical line, the region under the curve y equals X times square root one minus X square, where X is between zero and one. So that's the left hand side of the red curve here. So this is why equals ex Route one minus X squared from zero to one. And then we rotate this line around the Green Line. X equals one, and we could another graph from one to two, so we should determine which method we should use here. Or should we use the shell method or should we try in the washer method? So if I were to try to use the washer method, say something like that looks like a washer right there because the volume would be obtained by the washers moving vertically, we would have to express the curve by solving for X due to the fact that would be using D y. So you would have to solve for X. You'LL have many solutions, and then we would have to figure out what side corresponds to the left curve wish I had corresponds to the right curve, and that could be more tedious. And so in this case, let's proceed by using the shower method instead. So there's my show at some point X. So in this case, it was used the shell method, and in this case we could still use the X. That's an advantage. So we don't have to write this equation appear for why we do not have to soft Rex because of the DX. So in this case, we have the the volume to pie from a to be So in this case, zero one and then we need the radius of the show Time's FX. So this is the volume when using the show methods. So this is the radius of what's to note this green show that I wrote here the cylindrical cell Listen notice by CX, then our is the radius of CX. And so looking at the cylinder with the radius is just this distance from one toe X. So our is just equal to one minus X so we can plug that in and then we could replace here. F of X is why equals X times square root one minus X squared. And so we have to pie in a girl zero one one minus x This is R R R since the X and then f of x. Okay, so ours corresponding to one minus x and in f of x corresponds to x square root, one minus x word. So I'm running out of room here. Let's go to the next page. So picking up where we left off, we can go ahead and multiply out the one minus X So we have ex radical one minus X square minus was pulled his tube eye Actually let me go ahead and distributed to start with that So we distribute the one minus X to FX and then we write it is two. In a girls, they look very similar, but the difficulty there are a little different. For example, in this first inaugural, you can use a use up if you want. But for the second one, we you should use it tricks up view. So it's just not gonna work here for the second interval. So that's one reason to split them both up. If you like you sub If you prefer over, it shrinks up. Then this first integral we can go that method. So it's good indignities with different colors. Let's not this first integral in red. So taking you use up here we should take you to be the expression in the radical. Then we have. Do you equals negative two x t x. So do you over Negative too equals X t X And that's exactly what we wanted. We want this X the ekstrom also the limits of integration will change. So in this case, if X equals zero, which is our lower limit Oh, then you becomes one minus zero square eagles one. And if x equals one the upper limit, then you equals one minus one square equals zero. So we have switched the women's of integration and so this first in our world becomes to pie. Then we have a negative one half from this, do you? And now, because our limits of integration switched and a girl from one zero and then we have you to the one half power to you and we could go ahead and integrate this less than two steps here. First we see that we could cancel off the two and then we know from one of our properties Let me write up here in the top right in a row From a to B The left is equal to negative in general beat it eh Of f So this is one of the properties The first properties of the interval that we've learned So I'll use that here I'll swish the zero and the one that don't introduce a minus sign and then the minus will cancel with this minus sign So we have pi times positive one integral zero one You too the one half to you and then we could integrate this using the power rule and then we multiply by two thirds and then we're plugging in zero and one and I don't just give us two pi over three So that's the first part of our answer and I will compute the second part and then we'll put those answers together and then we'LL subtract So let's go on to the next page For the second interval Second inaugural appear rsr Interval was to pie Hey Sierra One ex cleared radical one minus x squares The ex hear that saying you sup freak is not going to work because of that square. In this case, I would do a regular tricks of you Could have been a trick Suffer the previous and overall. But use of this fine. But in this case, we should go ahead and take thanks to be signed data. So here is one. Remember when you do it, tricks up involving sign We have a restriction for data and here's a restriction. So we'LL have to keep that in mind when we want to come switch their primitive integration. And then here we have the X coz I next the XT, sir, not for the limits. Here we have zero. So plugging that in for X into this equation we have zero equals scientist And if they does in this interval appear we must have stated equal zero. So our lower limit will still be zero and then for the upper limit X is one. So we have one equal scientist, Ada And the only time that happens in our interval is at the end point pilots who So these are our new limits of integration. Let's go ahead and rewrite this in general. So this is all equal to We have to pi in a Girl in Sierra Power, too. And then we have X clear Solutions science, where that's directly from our drinks up here and then we have of radicals. Let's go to the side and simplify this radical one minus X squared is one minus sine squared data. This's co sign squared by the potato and identity. And finally, that's justcause data. So we have co Santana for the radical. But then also watch out for this d x. The DX also had co sign, so we'LL get another co sign factor. So one from this day and one from the DX Not for the next step. Let's use the fact that you have these half angle formulas for signing co sign. So those are the half Ingle identities and remote supplying these together that's going to put a foreign the denominator. So let's go ahead and pull out that for outside of the integral and then on the inside. We have one plus co sign to data tens one minus costs into data, so that becomes one minus co signed Square two there, let's go ahead and cancel the two's here. That, too, in the four so we have power too. And the girls there are no one and by the photographer identity again, we know that this is Science Square, but of two data. So we'LL use this half angle identity again, But this time we have to date it inside of the sign square. So when we multiply that by two on the right hand side, onboard data. So that's our next Step four. So we had signed square to data that becomes one minus co sign for theta over, too. And let's just go ahead and pull up that two out of the denominator with one minus co sign for data and I were ready to integrate. Excuse me, That should have been by over two. So now taking the integral we have power before and a girl of one is just data and in the role of co sign for data. It's signed for data for and her end points or zero firing two. And let's go ahead and plug those in. So the data's pie over too minus sign off to pie. Oh, and then plug in zero and then cancels much as we can sign it to hire zero sign of zero zero. There's another zero. So we're left over with pi over four times pie, too. Pi square number eight. So this is the second part of the answer. So taking both of our answers and combining these together we have two pirate three. This was the first in a role that when we did the use institution and then our newest intern girl that we were subtracting using the truth substitution, and there's a final answer.
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