How to evaluate $\iint_R \cos (\max \{x^3,y^{3/2}\}) dx dy$ , where $R:=[0,1]\times [0,1]$ ?

Question

How to evaluate $\iint_R \cos (\max \{x^3,y^{3/2}\}) dx dy$ , where $R:=[0,1]\times [0,1]$ ? I tried breaking the region to do case by case , but I am not getting anywhere . Please help . Thanks in advance

Answer 1

Note that, for $x$ and $y$ in $[0,1]$, $x^3\ge y^{3/2}$ iff $y\le x^2$ and $y^{3/2}\ge x^3$ iff $x\le\sqrt{y}$. Thus $$\iint_R\cos(\max\{x^3,y^{3/2}\})dxdy=\int_0^1\int_0^{x^2}\cos(x^3)dydx+\int_0^1\int_0^{\sqrt{y}}\cos(y^{3/2})dxdy$$