I have a question "Sketch the region of integration and, without evaluating, determine whether the integral is positive, negative, or zero. Then evaluate each integral."
$$\int_0^{\pi/2}\int_0^{\pi/2-x}\sin(x+y) \ dy\ dx$$
I can definitely evaluate this and it evaluates to $\int_0^{\pi/2}\int_0^{\pi/2-x}\sin(x+y) \ dy\ dx = 1$ which is a positive number.
The way this question is phrased makes it seems as though I have to determine if it is positive before I evaluate it. Can someone outline how I would do something like this?
The integral gives the signed area under the graph of a function. If the graph of the function is above the x-y plane (in other words, the function is positive over the region of integration) then the function will definitely have a positive integral.
All you need to do is sketch the parts of the plane where $\sin(x+y)$ is positive. If your region of integration falls inside one of those regions,then the integral will be positive.