Question

Where e is enclosed by the planes z = 0 and z = x + y + 9 and by the cylinders x2 + y2 = 4 and x2 + y2 = 16?

Answer 1

Best to use cylindrical coordinates. The volume (assuming that's what you're supposed to find) of the region is given by the triple integral


`\displaystyle\iiint_E\mathrm dV=\int_(\theta=0)^(\theta=2\pi)\int_(r=2)^(r=4)\int_(z=0)^(z=r(\cos\theta+\sin\theta)+9)r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta`

`=\displaystyle\int_(\theta=0)^(\theta=2\pi)\int_(r=2)^(r=4)(r^2(\cos\theta+\sin\theta)+9r)\,\mathrm dr\,\mathrm d\theta`

`=108\pi`