What are the common methods and tools to tackle optimization problemsinvolving integrals. To be precise lets consider the following optimization problem that I came across with: $$\text{maximize}\,\,F(a,b)=\int_0^a\int_0^bf(x,y)dxdy,$$ Subject to $ab=1$.
Any reference will be highly appreciated.
One useful tool is the Fundamental Theorem of Calculus, which helps you find the partial derivatives of $G(a,b) = \int_0^a \int_0^b F(x,y)\; dx\; dy$ with respect to $a$ and $b$. Of course, the result will still be in terms of integrals, so there will still be work to do.