How to find the area of region $R$? Region $R$ contains all the points $(x,y)$ such that $x^2+y^2\leq100\;$  and $\:\sin(x+y)\geq0$. Find the area of region $R$.

$\:\:\:\:\:\:\:\:\:$$\sin(x+y)\geq0$
$\:\:\:\:\:\:\:\:\:$$\implies2n\pi\leq x+y\leq(2n+1)\pi$

Thus graph would be probably look like this:

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$

Now how to find the area?

try to divide it to regains, start with the top right quarter, find the equations for the start and end of the region that $\sin(x+y)\geq0$ is true(where it is equal $0$)
 at them and then integrate them with each other, the axises and the circle of $r=10$. after that do the top left one. now if you want you can prove symmetry and just multiply by $2$ or keep going and integrate the bottom left and bottom right(i think keep integrating is better for practice)
