Having such integral, how to optimize it in maple?

Question

So we have :

 (1/3)*sig0*h^3*(int(int(sin((1/3)*arctan(y, x)), x = 0 .. r), y = 0 .. 2*Pi))

Is it possible to optimise it? (in maple or any other way...)

How I got here:

> R := 1; 
H := h;
sigma[0] := sig0; 
sigma := sigma[0]*z^2*sin((1/3)*`ϕ`); 
toCar := eval(sigma, {r = sqrt(x^2+y^2), z = z, `ϕ` = arctan(y, x)});

> Q := int(int(int(toCar, x = 0 .. r), y = 0 .. 2*Pi), z = 0 .. H);

Answer 1

By optimize, I assume you are meaning simplify. I don't see a parameter to optimize over. The limits on the integrals look like the differentials should be $\rho \; d\rho \; d\theta$, not $dx \; dy$. If so, you have $\int_0^{2\pi}\int_0^r\sin(\frac{\theta}{3})\rho \; d\rho \; d\theta=-3\frac{r^2}{2}\cos(\frac{\theta}{3})\mid_0^{2\pi}=\frac{9r^2}{4}$