Given $G=E\cap F$ with $$E=\{(x,y,z)\mid x^2+y^2+z^2\leq4z\}$$ and $$F=\{(x,y,z)\mid3x^2+3y^2\leq z^2\}$$
I have to evaluate: $$\iiint_{G}\exp(-\sqrt{(x^2+y^2+z^2)^3})dV$$
Using spherical coordinates, I got $$G=\{(\rho,\theta,\phi)\mid0\leq\theta\leq2\pi, 0\leq\phi\leq\pi/6, 0\leq\rho\leq4\cos\phi\}$$
However, since $x^2+y^2+z^2=\rho^2$, the first integral becomes $$\int_{0}^{4\cos\phi}\exp(-\rho^3)\rho^2d\rho=\frac{1}{3}(-\exp(-64\cos^3\phi)+1)$$ and then, the second integral is $$\int_{0}^{\pi/6}\frac{1}{3}(-\exp(-64\cos^3\phi)+1)\sin\phi d\phi$$
My question is: how do I evaluate (using elementary functions) the following integral $$\int\exp(64\cos^{3}\phi)\sin\phi d\phi$$
PS: I'd also appreciate if someone could tell me that everything is right.