evaluating integrals

Quote:

Originally Posted by calcbeg

Hi I am not sure what to do with this one

Evaluate the following integral

definite integral sign b="a" ; a=0 ( x )* ("a"^2 - x^2)^1/2 dx

where "a" is a positive constant

I know the x must be x^2 on the other side but I really don't know what to do with this mess.

Directions please

This calculus beginner

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

substitution ... let $u = a^2 - x^2$

$du = -2x \, dx$

$-\frac{1}{2} \int_0^a -2x\sqrt{a^2-x^2} \, dx$

substitute and reset the limits of integration ...

$-\frac{1}{2} \int_{a^2}^0 \sqrt{u} \, du$

$\frac{1}{2} \int_0^{a^2} \sqrt{u} \, du$

integrate and evaluate the last definite integral using the FTC

evaluating integrals - a and x