Question

Find the volume of the following solid. Using cylindrical coordinates
z = 4 - 4(x^2 + y^2)

z = (x^2 + y^2)^2 -1

Answer 1

In cylindrical coordinates, the equations of the surfaces become

$z = 4 - 4r^2 \\\\ z = (r^2)^2 - 1 = r^4 - 1$

These surfaces intersect on the cylinder of radius 1 with cross sections parallel to the x,y-plane:

$4 - 4r^2 = r^4 - 1 \\\\ \implies r^4 + 4r^2 - 5 = (r-1)(r+1)(r^2+5) = 0 \\\\ \implies r=1$

Then in cylindrical coordinates, the volume of the space bounded by these surfaces is

$\displaystyle \int_0^{2\pi}\int_0^1\int_{r^4-1}^{4-4r^2}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta = 2\pi \int_0^1 \int_{r^4 - 1}^{4-4r^2}r\,\mathrm dz\,\mathrm dr \\\\ = \pi \int_0^1 (4-4r^2)^2 - (r^4-1)^2 \,\mathrm dr \\\\ = \pi \int_0^1 (15-32r^2+18r^4-r^8)\,\mathrm dr = \boxed{\frac{352\pi}{45}}$