Question

This is a definite integral question.

Evaluate the following integral:

(0)S(a)((x)((a^2 - x^2)^(1/2)))dx

with a being a constant and the (0) being at the bottom of the integral notation and (a) at the top. S is the integral notation.

I firstly checked whether the function was even/odd. (x) is odd, and ((a^2 - x^2)^(1/2)) is even, therefore the question answers 0. I am hesitant as to whether this is right because the professor did not do this and went straight to solving the question using substitution (and the outcome was not 0.)

I appreciate any help.

Answer 1

let u = a^2-x^2 and you have
du = -2x dx
So the integral becomes

∫[a,0] -1/2 √u du

= -1/3 u^(3/2) [a^2,0]
= 1/3 a^3

Answer 2

Thanks Steve. Does that mean the function is not odd? I thought odd functions are always integrated to = 0 because of their symmetry.

Answer 3

the integral is 0 if the integration limits are -c and +c, since f(-c) = -f(c)

That is not the case here.

Answer 4

Ah Thanks!