Cylindrical coordinates: polar coordinates plus a height.

Find the volume of a cylinder of radius \(a\) and height \(h\).

1. \(\displaystyle\int_0^{2\pi}\!\!\int_0^a\!\!\int_0^h r\,dz\,dr\,d\theta = 2\pi\cdot\frac{a^2}{2}\cdot h\).

Answer\(\pi a^2 h\)

Find the volume bounded by the paraboloid \(z = x^2 + y^2\) and the plane \(z = 4\).

1. In cylindrical form the paraboloid is \(z = r^2\). It meets \(z = 4\) at \(r = 2\).
2. \(\displaystyle\int_0^{2\pi}\!\!\int_0^2\!\!\int_{r^2}^4 r\,dz\,dr\,d\theta = 2\pi\int_0^2 (4r - r^3)\,dr = 2\pi(8 - 4)\).

Answer\(8\pi\)

Find the volume of the cone \(z = \sqrt{x^2 + y^2}\) cut off by the plane \(z = h\).

1. The cone is \(z = r\), from \(z = r\) up to \(z = h\), for \(0 \le r \le h\).
2. \(\displaystyle 2\pi\int_0^h (h - r)\,r\,dr = 2\pi\left(\frac{h^3}{2} - \frac{h^3}{3}\right)\).

Answer\(\dfrac{\pi h^3}{3}\): a cone of radius \(h\) and height \(h\)

Evaluate \(\displaystyle\iiint (x^2 + y^2)\,dV\) over the cylinder \(x^2 + y^2 \le a^2\), \(0 \le z \le h\).

1. \(\displaystyle\int_0^{2\pi}\!\!\int_0^a\!\!\int_0^h r^2\cdot r\,dz\,dr\,d\theta = 2\pi\cdot\frac{a^4}{4}\cdot h\).

Answer\(\dfrac{\pi a^4 h}{2}\)

Evaluate \(\displaystyle\iiint z\,dV\) over the region between \(z = x^2 + y^2\) and \(z = 1\).

1. \(\displaystyle\int_0^{2\pi}\!\!\int_0^1\!\!\int_{r^2}^1 z\,r\,dz\,dr\,d\theta = 2\pi\int_0^1 \frac{r(1 - r^4)}{2}\,dr\).
2. \(\displaystyle\pi\left(\frac{1}{2} - \frac{1}{6}\right)\).

Answer\(\dfrac{\pi}{3}\)