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Problem 64 Hard Difficulty

Evaluate the definite integral.

$ \displaystyle \int^a_0 x\sqrt{a^2 - x^2} \, dx $


$\int_{0}^{a} x \sqrt{a^{2}-x^{2}} d x=\frac{1}{3} a^{3}$

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we know that Are you? Convene what's under the square root, which is a scored minus X squared which means r D X is now negative 1/2 do you over acts which means our bounds are now going to be from a squared to zero because we know a is a squared minus a score which is zero and be a zero squared plus a scored, which is simply just gonna give us a squared. Let's let us use the power rule, which means we increase the expert by one and we divide by the new exponents. Now we can simply plug in our bounds. This gives us 1/3 a cute