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JH
Numerade Educator

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Problem 21 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{0.6} \frac{x^2}{\sqrt{9 - 25x^2}}\ dx $

Answer

$$\frac{9}{500} \pi$$

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Video Transcript

Let's evaluate the integral from zero to zero point six of X square over the square root of nine minus twenty five x squared. So looking inside the radical in the denominator nine minus twenty five X squared weaken. Right, This is three squared, minus parentheses, five x squared. And now this looks very similar to a Z expression of the form A squared minus X where, and this is why we should do a substitution. Five X equals a scientist were raised three, Which means X is three signed data over five. That's our troops of and we have d X equals three co Santa over five data. Now, since this is a definite liberal, we could switch these limits of integration. So let's plug in X equals zero. I also noticed that for Atrix up involving Sign we always take satar to be betweennegative Kyra too. And clever too. So plugging in our lower bound X equals zero and for X, we have zero on the left and on the right. We have three over five signed data, which means signed data equals zero. And since data is in this interval, this means state. It has to be zero now for the upper bound plug in point six Rex. And on the right, we have three over five, which we could also write his point six signed data. So this means scientist Ada equals one. And the only time that happens in our honorable for data interval over here is that pirate too. So that's upper bound. Now let's rewrite this inner girl using data. So we have seen your apartment too. On the top, we have X squared. So that's nine over twenty five and then sine squared data. And then the ex is three over five coastline data. And in the denominator, we have nine minus twenty five x squared. So here we right, twenty five x squared equals five X squared equals three. Scientist A But But this is by the tricks up squared. So we have nine sine squared, all right. And the denominator weaken Pull out and nine outside the radical that becomes a three. We also have twenty five times five And in the numerator we have nine times three and left over in the bottom. We have one minus sign square Now by the protagonist identity. We know that we can right this as radical co signs for data The nationals co Santa Then we could go ahead and cancel coastlines. We'LL have a co sign in the denominator and we also have one in the numerator, so we could cross those off and we're left with nine over one twenty five cancelling those threes. Science for data Co signed it sees me, we drop. Sorry about that. We cancel the coastline, so we just have science for data. So Mi re write this on the next page. Since I'm running out of room, this is what we had in the previous expression. And then now this is a trick in a girl we have signed squared. We used a half angle formula to rewrite this as one minus coastline to data. Almost so. If we do this, we have nine over one twenty five and then we have one minus cosign to date over too, which we could hand over it. Nine over one twenty five. State over to minus sign tooth aito over four and then forget those end points. You're a pirate. So if we plug in faded equals pi over too. That power for minus sign pi over four, and then when we plug in zero, we have zero over too, minus sign zero over. For now, sign a pie. A zero zero and sign of zero is also zero. So we're just left with nine over one twenty five Time's pyro, for which we can raise nine pie over five hundred. Oops, and there's a final answer.