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

Explanation:

Answer 2

Using cylindrical coordinates,


`\displaystyle\int_(x=0)^(x=\sqrt3)\int_(y=0)^(y=√(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`

`=\frac{9\pi}4`