Question

Why is the integral of an odd function from -A to A equal to zero?
The definition of an odd function is f(-x)=-f(x)
Taking the Integral from -A to A, we should obtain F(A) - F(-A)
plugging in what we know from the properties of an odd function:
F(A) - (-F(A)) = 2F(A)
but from all my examples in class and even wikipedia, it says that I should getting 0.

Answer 1

because the function has rotational symmetry about the origin and is above and below the x-axis.
The area above and the area below will have the same value but opposite signs.. hence the zero result.

To avoid it you need to find the point where the curve cuts the x axis call it B

and

\[\left| \int\limits_{-A}^{B} f(x) dx \right| + \int\limits_{B}^{A} f(x) dx \]

which then becomes 2F(A)

Answer 2

Just because a function is odd, doesn't mean the antiderivative is odd. You seems to be using the odd property on the antiderivative (when you do F(-A)=-F(A)), but that is not true.