# Triple integrals in cylindrical coordinates

### Triple integrals in cylindrical coordinates

#### Lessons

Notes:

Recall that when converting from Cartesian Coordinates to Polar Coordinates with double integrals we do the following:

$\int \int_D f(x,y)dA = \int^{\theta=\beta}_{\theta=\alpha} \int^{r=g_2(\theta)}_{r=g_1(\theta)} f(r \cos \theta, r \sin \theta ) r dr d\theta$

With triple integrals, it is very similar. Instead of calling it polar coordinates, we call it cylindrical coordinates.

Suppose we want to triple integrate $f(x,y,z)$ in cylindrical coordinates in the following region of $E$.

$\alpha \leq \theta \leq \beta$
$g_1 (\theta) \leq r \leq g_2 (\theta)$
$h_1 (r \cos \theta, r \sin \theta) \leq z \leq h_2(r \cos \theta, r \sin \theta)$

Then

$\int \int \int_E f(x,y,z) dV = \int^{\beta}_{\alpha} \int^{g_2(\theta )}_{g_1(\theta )} \int^{h_2 (r \cos \theta, r \sin \theta)}_{h_1 (r \cos \theta, r \sin \theta)} f(r \cos \theta, r \sin \theta, z) rdzdrd\theta$

• Introduction
Triple Integrals in Cylindrical Coordinates Overview:
a)
Triple Integrals in Cylindrical Coordinates
• Polar Coordinates $\to$ Cylindrical Coordinates
• All $x$'s & $y$'s change to $r$'s & $\theta$
• $z$ stays the same

b)
An Example of Converting to Cylindrical Coordinates
• All $x$'s & $y$'s change to $r$'s & $\theta$
• The variable $z$ stays the same
• Add an extra $r$
• Integrate

• 1.
Converting to Cylindrical Coordinates
Convert the following triple integral to cylindrical coordinates

$\large \int_{-2}^{0}\int_{0}^{\sqrt{4 - x^{2}}} \int_{2x^{2} + 2y^{2}-4}^{x+ y} \sqrt{9 - x^{2} - y^{2}}\, dz \,dy\, dx$

• 2.
Convert the following triple integral to cylindrical coordinates

$\large \int_{-3}^{3}\int_{-\sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} \int_{3z- 5}^{2} \, zIn(x^{2} + y^{2})\, dx \,dz\, dy$

• 3.
Converting & Integrating
Evaluate $\, \int\int\int_{E} 2dV \,$ where $\, E \,$ is the region bounded by $\, z = x^{2} + y^{2} - 2 \,$ and $\, z = 6 - x^{2} - y^{2}$.

• 4.
Evaluate $\, \int\int\int_{E} 1dV \,$ where $\, E \,$ is the region bounded by $\, z = 4, z = x - y - 3 \,$ and inside $\, x^{2} + y^{2} = 4$.