The question says to sketch the region of integration and reverse the order of integration and then evaluate if possible.
$$\int_{0}^{\sqrt{3}}\int_{0}^{\tan^{-1}(y)}\sqrt{xy}dxdy$$
This is what I graphed as the region of integration:
So when I reverse the order of integration I get:
$$\int_{0}^{\tan^{-1}(\sqrt{3})}\int_{\tan x}^{\sqrt{3}} \sqrt{xy} dy dx$$
$$=\int_{0}^{\tan^{-1}(\sqrt{3})} \sqrt{x}\frac{2}{3}y^{3/2} \Bigg|_{\tan x}^{\sqrt{3}} dx$$
$$=\int_{0}^{\tan^{-1}(\sqrt{3})} \frac{2}{3}\sqrt{x} \Bigg[3^{3/4}-(\tan x)^{3/2}\Bigg]dx$$
$$=\frac{2}{3}\int_{0}^{\tan^{-1}(\sqrt{3})} 3^{3/4}\sqrt{x}-\sqrt{x}(\tan x)^{3/2}dx$$
I don't know what to do next to evaluate the integral. Can someone please help me? The question says to find the value of the integral if possible, so does that mean I cannot find the value of this integral? If so, why not?
