Question

Use polar coordinates to find the volume of the given solid. bounded by the paraboloid z = 5 + 2x2 + 2y2 and the plane z = 11 in the first octant.

Answer 1

Using cylindrical coordinates,

$\displaystyle\int_{x=0}^{x=\sqrt3}\int_{y=0}^{y=\sqrt{3-x^2}}\int_{z=5+2x^2+2y^2}^{z=11}\mathrm dz\,\mathrm dy\,\mathrm dx$
$\displaystyle=\int_{\theta=0}^{\theta=\pi/2}\int_{r=0}^{r=\sqrt3}\int_{z=5+2r^2}^{z=11}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\frac\pi2\int_{r=0}^{r=\sqrt3}r(11-(5+2r^2))\,\mathrm dr$
$=\displaystyle\pi\int_{r=0}^{r=\sqrt3}(3r-r^3)\,\mathrm dr$
$=\dfrac{9\pi}4$