Integral by Trig Substitution, Calc 2

[Darkestsolrac]

Homework Statement

The definite integral of ∫(x^2 √(a^2-x^2) dx from 0 to a

Homework Equations

The Attempt at a Solution

So i don't need actual help with this problem. I got the answer, (π*a^4)/16 and I verified with the back of the book. The question I have is whether this problem merits an entire side of work? None of the examples my professor has given have ever been more than a few lines of work and this took me a whole side of a paper. Am I being inefficient or should I just expect this from now on?

Oh and sorry if my notation bad or if this should be on another thread, this is my first post lol .__.

 

[LCKurtz]

I guess it amounts to how much you write. It requires a trig substitution followed by a double angle formula. Is that how you did it? Did you carry the limits along with the substitution or did you back substitute to x? That takes more steps. I used about 1/2 of one side of a standard sheet of paper for it.

 

[Darkestsolrac]

I decided to back substitute into x, i thought converting the limits would be a hassle on this question. I think my issue was getting a sin4θ and not knowing any quick identities to simplify. Oh well, thanks for the response

 

[ehild]

You could have changed the integration limits after substitution. If you substituted x/a=sin(u) then the integral with respect to u goes from 0 to pi/2.

ehild