I want to convert integrals $\displaystyle \int_0^{\frac{\pi}{2}} \int_1^{\sqrt{3}} \int_1^{\sqrt{4 - r^2}}\,\, r^3(\sin(\theta) \cos(\theta))z^2\,\, dz\,\, dr\,\, d\theta$ from cylindrical coordinate

to rectangular coordinate($\displaystyle x,y$ and $\displaystyle z$). Is there an easy way to convert these integrals from cylindrical

coordinate to rectangular coordinate?

What is the best method for this conversion?

The answer for the above problem is: $\displaystyle \int_0^1 \int_{\sqrt{1-x^2}}^{\sqrt{3-x^2}} \int_1^{\sqrt{4 - x^2 -y^2}}\,\, z^2yx\,\, dz\,\, dy\,\, dx + \int_1^{\sqrt{3}} \int_0^{\sqrt{3-x^2}} \int_1^{\sqrt{4 - x^2 -y^2}}\,\, z^2yx\,\, dz\,\, dy\,\, dx$

Thanks.