Simple integration question?

Question

How do you integrate the following:

The definite integration from 0 ==> r ( 0 to "r" being the limits of the integral ) of :

x ( r^2 - x^2 ) ^ (1/2) dx

[ the square root of ( r^2 - x^2 ) ]

Thanks.

Answer 1

Let u = r^2 - x^2    (if x = 0, u = r^2   and if x = r, u = 0

du = -2x dx

INT ( x(r^2 - x^2)^(1/2) dx, 0, r)

= (-1/2) INT ( (r^2 - x^2)^(1/2)(-2x)dx, 0, r)

= (-1/2) INT ( u^(1/2) du, r^2, 0)

= (-1/2) (2/3) u^(3/2) eval from r^2 to 0

= (-1/3) ( 0^(3/2) - (r^2)^(3/2) )

= (1/3)r^3

Hope that helps.

--David