Originally Posted by

**oblixps** the definition of an odd function is f(-x) = f(x). if you plug in -y into (y^2)siny, you get (y^2)sin(-y) and since sin is an odd function itself, sin(-y) = -sin(y) so (y^2)sin(-y) = -(y^2)siny and this shows that (y^2)siny is an odd function.

also from the definition of an odd function f(-x) = f(x), you can see that all odd functions are symmetric with respect to the origin. if you graph this function out notice that half the function is above the x axis and half is below the x axis. since you are integrating over a symmetric interval (from -4 to 4) both regions (from -4 to 0) and (from 0 to 4) are equal so when you add the two regions together you get zero. the integral is just a way to sum up an infinite number of objects so if you are adding negative objects to the positive objects of the same magnitude, they will cancel out to 0.