Question

Can someone evaluate this definite integral, please?

The definite integral of x((a^2)-(x^2))^(1/2): Upper limit = a, Lower limit = 0.

Answer 1

Here's an image of the equation!

Answer 2

All you have to do is use substitution. \[\int\limits_{0}^{a}x*\sqrt{a^2-x^2}dx=-1/2*\int\limits_{\sqrt{a}}^{0}\sqrt{u}du=-\sqrt{a^2-a^2}^{3}/3+\sqrt{a^2-0^2}^{3}/3\]

Thus, the definite integral is equal to \[(a ^{2})^{3/2}/3\]

Answer 3

a simple u-substitution will work nicely...

u = a^2 - x^2
du = -2xdx ---> \(\large -\frac{1}{2}du=xdx \)