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Evaluate the integral.

$ \displaystyle \int_0^a x^2 \sqrt{a^2 - x^2}\ dx $

$$\frac{\pi}{16} a^{4}$$

Integration Techniques

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Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

Let's evaluate the integral from zero a of X squared times the square root of a squared minus X wears now inside the radical we have a squared minus X square. So our tricks up should be X equals a sign data from which we have DX equals Any co sign Taito D data. Since this is a definite or rule, we should see if we could change the limits of integration in terms of the new variable data. So using our truth substitution, plugging and X equals zero. Yeah, zero equals scientific data. Excuse me? A signed data. And also, when we do it in a girl, a trick some of this form. It will always put the condition that data's betweennegative over too, and powerful too. So this is for the a sign data tricks up. So here we have a signed data equals zero. So assuming that is down zero, we have signed equal zero, and the only time that happens in this honorable zero. So that's our new lower limit for the upper limit. We have a so plugging in a for X with a equals a signed era, so divide both sides by a Scient ADA equals one, and the only time that happens in our interval is that over too. So those are new limits of integration and zero and power too. So before we agree, right this original problem, let's just simplify this term right here. Square root of a squared minus X squared. So that's a square science. Where data here I could pull out of a squared or left over with one minus sign square. So we had eight times Square root co sign squared data equals Saeko San Diego. So plugging all this in we could write Original problem is the inaugural from zero to pie over too. X squared becomes a squared sine data sine squared data and then the radical. We just simplified this. This's a co sign data and then DX was also a co santa. So here, let's come. Let's simplify this So we have AIDS to the fourth power. Let's pull that constant outside the integral. And then we have science where times co sign squared. Now this is a trick in a rule, and since we have even powers on both side and co sign will want to use the Haft angle identities, science square is one minus coastline to data over to and co sign squared one plus coastline to date over, too. So let me go to the next page, since I'm running out of room so multiplying those two's together we get a four on the bottom and then one minus co sign to data one plus goes into We could re simply by this angel. Fourth over four one minus co sign squared potato And then using the Pythagorean identity, we can write this assign squared two There now, once again using the half angle formula Weaken right sine squared to data is one minus co sign of two times to Daito. So it's a fourth Ada over soon, and I should be consistent. This's a definite liberal and from the previous page our end points for See you in a pile over too. So we still need those now replacing sine squared with this one minus co sign Fourth eight over, too. Went into the fourth over eight in a girl C on a paper too one minus coast. And for data And this we know how to number eight. This is eight of the fourth over eight in a roll of one is data in a World Co sign for theta designed for theta over four. So now we plug in the end, points into this expression went into the fourth over eight pie or two minus sign of four times Pirates Who it's to pie. And then when we plug in zero, we have zero minus sign zero over to over four. So these terms are both zero sign of two pi zero. So we're left over with a the fourth over eight times. Pyro, too, which we can write his pie into the fourth over sixteen, and there's our final answer.