Consider the regions $A = \{(x,y) | x^2 + y^2 \leq 100\}$ and $B = \{(x,y) | sin(x+y) > 0\}$ in the x-y plane.
Find area of the region $A\cap B$ .
The area A is the inside area of the circle $x^2 + y^2 = 100$ while region B is many regions represented by $ 0 \lt x+y \lt \pi $ and $ 2\pi \lt x+y \lt 3\pi $ and so on. These just are a bunch of straight lines and the whole thing is quite easy to plot. (Like I do here)
We were supposed to solve this in under 5 minutes. One method would be to find the intersection of each of the lines with the circle and use geometry or integration then add all of them. This is quite elaborate and tedious. If you see my plot linked above 5 regions coincide. I was wondering if there is an alternate and more easy method to solve this. I sense some shifting and rotating cutting and pasting might make this problem a little more easy to solve.
By symmetry the area is half the area of the circle that is $50\pi$.
Indeed for the strip $0 \lt x+y \lt \pi$ we have that $\sin(x+y)>0$ and for $-\pi \lt x+y \lt 0$ $\sin(x+y)<0$ and so on, thus the strip with $\sin(x+y)>0$ cover an half of any region symmetric with respect to the line $y=-x$.