# What will be the volume of the cylinder: x^2 + y^2 = 4 and the planes y+z=4 and z=0?

May 18, 2018

$16 \pi$

#### Explanation:

Cylindrical Volume:

$V = {\int}_{V} r \setminus \mathrm{dr} \setminus \mathrm{dz} \setminus d \theta$

$= {\int}_{\theta = 0}^{2 \pi} {\int}_{r = 0}^{2} {\int}_{z = 0}^{4 - r \sin \theta} \setminus r \setminus \mathrm{dz} \setminus \mathrm{dr} \setminus d \theta$

$= {\int}_{\theta = 0}^{2 \pi} {\int}_{r = 0}^{2} \setminus {\left(r z\right)}_{z = 0}^{4 - r \sin \theta} \setminus \mathrm{dr} \setminus \setminus d \theta$

$= {\int}_{\theta = 0}^{2 \pi} {\int}_{r = 0}^{2} \setminus 4 r - {r}^{2} \sin \theta \setminus \mathrm{dr} \setminus \setminus d \theta$

$= {\int}_{\theta = 0}^{2 \pi} {\left(\setminus 2 {r}^{2} - {r}^{3} / 3 \sin \theta\right)}_{r = 0}^{2} \setminus \setminus d \theta$

$= {\int}_{\theta = 0}^{2 \pi} \setminus 8 - \frac{8}{3} \sin \theta \setminus \setminus d \theta$

$= {\left(\setminus 8 \theta + \frac{8}{3} \cos \theta \setminus\right)}_{\theta = 0}^{2 \pi}$

$= 16 \pi$

This is same as:

• $\pi \left({2}^{2}\right) \cdot 2 + \pi \left({2}^{2}\right) \cdot \frac{6 - 2}{2} = 16 \pi$

...which you get if you just splice and dice the cylinder using symmetry