Section 4.5 Cylindrical Integral

4.5 Cylindrical Integral

Recall the Cylindrical Coordinates system is denote a point in a 3-dimensional space via $(r,\theta ,z)$ where $(r,\theta )$ is the polar coordinates of the point project onto the $xy$-plane. Hence we have relationships,

$$x^2+y^2=r^2,x=r\text{cos}(\theta ),y=r\text{sin}(\theta ),z=z.$$

Triple integrals can often be more readily evaluated by using cylindrical coordinates instead of rectangular coordinates. Some common equations

of surfaces in rectangular coordinates along with corresponding equations in cylindrical coordinates are 

Circular cylinder:  $x^2+y^2=r^2;$

Circular cone:  $z^2=x^2+y^2;$

Sphere: $x^2+y^2+z^2=c^2;$

Paraboloid:  $z=x^2+y^2.$

Definition: Triple Integrals

Consider the cylindrical box (expressed in cylindrical coordinates)

$$B=\{(r,\theta ,z)|a\le r\le b,\alpha \le \theta \le \beta ,c\le z\le d\}.$$

If the function $f(r,\theta ,z)$ is continuous on $B$ and if $(r_{ijk}^{\ast },{\theta }_{ijk}^{\ast },z_{ijk}^{\ast })$ is any sample point in the cylindrical $B_{ijk}=[r_{i-1},r_i]\times [{\theta }_{j-1},{\theta }_j]\times [z_{k-1},z_k]$, then we can define the triple integral in cylindrical coordinates as the limit of a triple Riemann sum, provided the following limit exists

$$\underset{l,m,n\to \mathrm{\infty }}{lim}\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(r_{ijk}^{\ast },{\theta }_{ijk}^{\ast },z_{ijk}^{\ast })r_{ijk}^{\ast }\mathrm{△}r\mathrm{△}\theta \mathrm{△}z$$

 if the limit exists.

Theorem: Fubini’s Theorem in Cylindrical Coordinates

Suppose that $g(x,y,z)$ is continuous on a rectangular box B, which when described in cylindrical coordinates looks like $B=\{(r,\theta ,z)|a\le r\le b,\alpha \le \theta \le \beta ,c\le z\le d\}$. Then $g(x,y,z)=g(r\text{cos}(\theta ),r\text{sin}(\theta ),z)=f(r,\theta ,z)$

and

$$\int \int {\int }_{B}g(x,y,z)dV={\int }_c^d{\int }_{\alpha }^{\beta }{\int }_a^{b}f(r,\theta ,z)rdrd\theta dz.$$

The iterated integral may be replaced equivalently by any one of the other five iterated integrals obtained by integrating with respect to the three variables in other orders. 

Example 1: Evaluating the triple integral $\int \int {\int }_B(zr\text{cos}(\theta ))rdrd\theta dz$ where $B=\{(r,\theta ,z)|0\le r\le 2,\frac{\pi }{2}\le \theta \le \pi ,0\le z\le 3\}$. 

Exercise 1: Evaluating the triple integral $\int \int {\int }_B(z\text{sin}(\theta ))rdrd\theta dz$ where $B=\{(r,\theta ,z)|0\le r\le 1,0\le \theta \le \pi ,0\le z\le 2\}$. 

Remark (Theorem): As before, we can do the integration over a general box, not just a cylinder one. Let $B=\{(r,\theta ,z)|g_1(\theta )\le r\le g_2(\theta ),\alpha \le \theta \le \beta ,u_1(r,\theta )\le z\le u_2(r,\theta )\}$ and $g(x,y,z)=g(r\text{cos}(\theta ),r\text{sin}(\theta ),z)=f(r,\theta ,z)$ be a continuous function over $B$, then 

$$\int \int {\int }_{B}g(x,y,z)dV=\int \int {\int }_{B}f(r,\theta ,z)rdzdrd\theta$$

$$={\int }_{\alpha }^{\beta }{\int }_{g_1(\theta )}^{g_2(\theta )}{\int }_{u_1(r,\theta )}^{u_2(r,\theta )}f(r,\theta ,z)rdzdrd\theta$$

Example 2: Let $E$ be the region bounded above by the cone $z=2-\sqrt{x^2+y^2}$ and below by the paraboloid $z=x^2+y^2$ . Set up a triple integral in cylindrical coordinates to find the volume of the region.

Exercise 2: Let $E$ be the region bounded below by the cone $z=2\sqrt{x^2+y^2}$ and above by the paraboloid $z=3-x^2-y^2$ . Set up a triple integral in cylindrical coordinates to find the volume of the region.

Example 3: Sketch the solid whose volume is given by the integral and evaluate the integral ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\int }_0^2{\int }_0^{r^2}rdzdrd\theta$.

Exercise 3: Sketch the solid whose volume is given by the integral and evaluate the integral ${\int }_0^{\frac{\pi }{2}}{\int }_0^1{\int }_{-r}^{0}rdzdrd\theta$.

Example 4: Sketch the solid. Evaluating the triple integral $\int \int {\int }_{E}xdV$ where $E$ lies between $x^2+y^2=1$, $x^2+y^2=4$ above the $xy$-plane below $z=x+2$. 

Exercise 4: Sketch the solid. Evaluating the triple integral $\int \int {\int }_{E}ydV$ where $E$ lies between $x^2+y^2=2$, $x^2+y^2=3$ above the $xy$-plane below $z=y$. 

Example 5: Sketch the solid and find its volume. The solid is bounded left by $xz$-plane, right by $x^2+y^2+z^2=4$, and outside the cylinder $x^2+z^2=1$. 

Exercise 5: Sketch the solid and find its volume. The solid is bounded below by $xy$-plane, above by $x^2+y^2+z^2=9$, and outside the cylinder $x^2+y^2=4$. 

Example 6: Sketch the solid and find the volume of the solid lies between the paraboloid $x=24-y^2-z^2$ and the cone $x=2\sqrt{y^2+z^2}$.

Exercise 6: Sketch the solid and find the volume of the solid lies between the paraboloid $y=x^2+z^2$ and the cone $y=4-3\sqrt{x^2+z^2}$. 

Example 7: Sketch the solid and find its volume. The solid is inside the sphere $x^2+y^2+z^2=2$, and the paraboloid $z=x^2+y^2$. 

Example 8: Change the integration into cylindrical coordinates

$${\int }_{-2}^2{\int }_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}{\int }_{\sqrt{x^2+y^2}}^{2}xzdzdxdy$$

Group work:

1. Sketch the solid and find its volume. The solid is inside the sphere $x^2+y^2+z^2=4$ and the cone $y=\sqrt{x^2+z^2}$. 

2. Change the integration into cylindrical coordinates

$${\int }_{-4}^4{\int }_0^{\sqrt{16-x^2}}{\int }_0^{16-x^2-y^2}\sqrt{x^2+y^2}dzdydx$$

3. Sketch the solid and find its volume. The solid lies inside both

the sphere $x^2+y^2+z^2=4$ and the cylinder $z^2+y^2=1$.