From my calc book:

"Since the integrand is an odd function, integral_(-2)^(2) (x^3)*(16-x^2)^1/2 dx = 0"

My question: Why?!

Question

From my calc book:

"Since the integrand is an odd function, integral_(-2)^(2) (x^3)*(16-x^2)^1/2 dx = 0"

My question: Why?!

anonymous · Answer

$$\int\limits_{-2}^{2}x^3\sqrt{16-x^2}dx = 0$$  <-----A little easier to view

anonymous · Answer

Also, I understand the trig substitution for the indefinite integral but I don't know why this definite integral is 0 and what an odd function is referring to.

anonymous · Answer

odd function is because the highest degree of x is 3

anonymous · Answer

and it is equal to 0 since the are from -2 to 0 "cancels" out the area from 0 to 2

anonymous · Answer

Will this usually happen for odd functions where it will "cancel" out?

anonymous · Answer

no unless it's an improper integral