Given the region
Calculate the integral
In a solution I have for this exercise it 's said that and that .
My question is, why is that this integral equals to zero? (there's no explanation about that in the solution I have)
Thanks for your reply. However, there are some things I haven't quite got (yet, I hope).
1. Why is odd, only because is odd?
2. Why does the integral from -4 to 0 cancels the integral from 0 to 4?
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.
The function goes below the x-axis and its integral from -2 to 2 is .
What we are saying is that if a function has a section above the x-axis and a section below the x-axis, and they are both the same "size", then part above the x axis will be positive, the part below the x- axis will be negative and they will cancel.
That is very different from just "having a line below the x-axis". The parts above and below the x-axis must be exactly the same to cancel.