When you get your new integral in the theta domain, what is the expression for that?
How do you solve this integral: From 0 to a, x^2*sqrt(a^2-x^2)
The answer is (1/16)a^4pi
I have no idea where pi comes from, so here's what I do
Let x=a*sin(theta)
My work ends up to be (4a^4/32) - (a^4sin4a/32)
Help!
Unfortunately, you error occurs somewhere in here- in exactly the part you did not show. Did you change the limits of integration as you worked? If x= a sin(theta), then when x= 0, a sin(theta)= 0 so theta= 0. When x= a, a= a sin(theta) so sin(theta)= 1, theta= pi/2.
My work ends up to be (4a^4/32) - (a^4sin4a/32)
Help!
Okay, thanks guys. I've figured out the problem, which is, as HallsofIvy stated, I didn't change my limits of integration in which 'a' should have been 'pi/2.' Could anyone possibly explain why if x=a, then theta=pi/2. I don't understand the logic behind that.
When you do a substitution on a definite integral, you must change over three things: the limits, the integrand, and the differential. For an indefinite integral, of course, while you don't need the limits, you must still transform the integrand and the differential.