Section 4.3 Double Integrals Polar

4.3 Double Integrals Polar

In previous section, we learn the double integral over general regions using rectangular coordinates $(x,y)$. This is in general not very practical if we have a region that is circular, for example a circle. It is naturally that we use the polar coordinate for the integration as we did in calculus II. Recall that any point $(x,y)$ in the plane, we can convert it into $(r,\theta )$ where $x^2+y^2=r^2$, $x=r\text{cos}(\theta )$ and $y=\text{sin}(\theta )$. Given

$$R=\{(r,\theta )|a\le r\le b,\alpha \le \theta \le \beta \},$$

 a polar region, one can find the volume between a continuous function $f(x,y)\ge 0$ and $xy$-plane with base $R$ using the polar coordinate system. 

Definition: Polar Coordinates in a Double Integral

The double integral of the function $f(r,\theta )$ over the polar rectangular region $R$ in the $r\theta$ -plane is defined as

$$\int {\int }_{R}f(r,\theta )dA=\underset{m,n\to \mathrm{\infty }}{lim}\sum _{i=1}^m\sum _{j=1}^{n}f(r_{ij}^{\ast },{\theta }_{ij}^{\ast })\mathrm{△}A$$

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

 $$={\int }_{{\theta }_1}^{{\theta }_2}{\int }_{r_1}^{r_2}f(r,\theta )rdrd\theta .$$

 When the function $f$ is given in terms of $x$ and $y$, using $x=r\text{cos}(\theta )$ and $y=\text{sin}(\theta )$, and $dA=rdrd\theta$ changes it to 

$$\int {\int }_{R}f(x,y)dA={\int }_{{\theta }_1}^{{\theta }_2}{\int }_{r_1}^{r_2}f(r\text{cos}(\theta ),r\text{sin}(\theta ))rdrd\theta .$$

Example 1: Compute the double integral ${\iint }_{R}f(x,y)dA$ where $f(x,y)=y$ and $R=\{(r,\theta )|1\le r\le 2,0\le \theta \le \frac{\pi }{2}\}$ 

Exercise 1: Compute the double integral ${\iint }_{R}f(x,y)dA$ where $f(x,y)=x$ and $R=\{(r,\theta )|2\le r\le 3,\frac{\pi }{3}\le \theta \le \frac{3\pi }{4}\}$ 

Example 2: Compute the double integral ${\iint }_{R}1-x^2-y^{2}dA$ where $R$ is the unit circle on the $xy$-plane. 

Exercise 2: Compute the double integral ${\iint }_R{x}^2+y^{2}dA$ where $R$ is a circle with radius $2$ on the $xy$-plane. 

Theorem: Double Integrals over General Polar Regions 

If $f(r,\theta )$ is continuous on a general polar region

$$R=\{(r,\theta )|h_1(\theta )\le r\le h_2(\theta ),\alpha \le \theta \le \beta \},$$

then

$$\int {\int }_{R}f(r,\theta )dA={\int }_{\alpha }^{\beta }{\int }_{h_1(\theta )}^{h_2(\theta )}f(r,\theta )rdrd\theta .$$

Example 3: Compute the double integral ${\iint }_{R}1dA$ where $R=\{(r,\theta )|0\le r\le 3\text{cos}(\theta ),\frac{\pi }{3}\le \theta \le \frac{2\pi }{3}\}$.

Exercise 3: Compute the double integral ${\iint }_{R}2dA$ where $R=\{(r,\theta )|0\le r\le \text{cos}(\theta ),\frac{\pi }{4}\le \theta \le \frac{3\pi }{4}\}$.

Example 4: Compute the double integral ${\iint }_{R}x+2ydA$ where $R$ is in the second quadrant enclosed by $x^2+y^2=9$, $x=0$ and $y=-x$. 

Exercise 4: Compute the double integral ${\iint }_{R}2x-ydA$ where $R$ is in the first quadrant enclosed by $x^2+y^2=4$, $y=0$ and $y=x$. 

Example 5: Compute the double integral ${\iint }_R{e}^{-x^2-y^2}dA$ where $R$ is in the third quadrant between the circles with center on the origin and radii $2$ and $3$. 

Exercise 5: Compute the double integral ${\iint }_R{e}^{x^2+y^2}dA$ where $R$ is in the second quadrant between the circles with center on the origin and radii $1$ and $2$. 

Example 6: Find the volume under the paraboloid $z=x^2+y^2$ and above $9\le x^2+y^2\le 25$. 

Exercises 6: Find the volume under the paraboloid $z=-x^2-y^2+4$ and above $1\le x^2+y^2\le 4$. 

Example 7: Evaluate 

$${\int }_0^2{\int }_0^{\sqrt{4-x^2}}{e}^{-x^2-y^2}dydx$$

 where $0\le y\le \sqrt{4-x^2}$ and $0\le x\le 2$.

Example 8: Find the volume of the solid above the cone $z=\sqrt{x^2+y^2}$ and below the sphere $x^2+y^2+z^2=1$.

Group work:

1. Evaluate 

$${\int }_0^3{\int }_0^{\sqrt{9-y^2}}{e}^{x^2+y^2}dxdy$$

 where $0\le x\le \sqrt{9-y^2}$ and $0\le x\le 1$.

2. Find the volume of the solid bounded by paraboloid $z=6-x^2-y^2$ and $z=x^2+y^2$.

3. Find the volume of the solid inside $x^2+y^2+z^2=25$ and outside the cylinder $x^2+y^2=9$.

4. Find the volume of the solid bounded by paraboloid $z=1+x^2+y^2$ and $z=5$.