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Problem 6 Easy Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^2 x^2 \sqrt{4 - x^2}\ dx $

Answer

$\pi$

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Video Transcript

Okay, so this question wants us to answer this integral using a tape. So if we look at our table, we find this formula, which is very close to what we're given. So we just want to transform our integral into something that looks like this. So, as you can see, there is nothing to do to this. It's already exactly in this form, we just see that you equals X and A equals two. So this means that d U equals DX. So you don't have to worry about changing our limits or any general factors. And we also know our values. A So that means we can just integrate. So are integral is equal to plugging into our formula. Here you, over eight times to you squared minus a squared, which is four times the square root of you squared minus four plus ace eight of the fourth over eight, which is 16 over eight. Sign in verse of you over, too. And we're going from zero two. So if we plugger limits in here, we get to over a times a minus four times square root of four minus four plus to sign in verse of to over two. So that's our plus limit. And then are negative. Limit turns into zero plus to sign in verse of zero, which is also zero. So that means our answer just becomes to sign in verse of one, which is two times pi over two or pie.

University of Michigan - Ann Arbor
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