Areas and Lengths in Polar Coordinates

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 10.4 – Areas and Lengths in Polar Coordinates

Expected Educational Results

• Objective 17–01: I can set up the integral to compute the area bounded by one or two polar curves.

• Objective 17–02: I can evaluate the integral to compute the area bounded by one or two polar curves.

• Objective 17–03: I can set up the integral to compute the arclength of a polar curve.

• Objective 17–04: I can estimate the integral for the arclength of a polar curve.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Areas and Lengths in Polar Coordinates

Area

Definition: Area Bounded by a Polar Curve

${\displaystyle A=\frac{1}{2}{\int }_a^b{r}^{2}d\theta }$$\displaystyle A=\frac{1}{2}\int_a^b{r^2\,d\theta}$

General Idea Behind Area Formula

In the Figure above, the area of the large circle is $A_{\text{circle}}=\pi r^2$$A_{\text{circle}}=\pi r^2$. The green sector is a fraction (with respect to the angles, i.e., area defined by $\theta$$\theta$ out of the area of the entire circle defined by $2\pi$$2\pi$, i.e., ${\displaystyle \frac{\theta }{2\pi }}$$\dfrac{\theta}{2\pi}$) of the whole circle. So

$\begin{array}{c}{A}_{\text{sector}}=\left({\displaystyle \frac{\theta }{2\pi }}\right)\pi r^2\\ =\left({\displaystyle \frac{\theta }{2\cancel{\pi }}}\right)\cancel{\pi }{r}^2\\ ={\displaystyle \frac{1}{2}}{r}^2\theta \end{array}$$A_{\text{sector}}=\left(\dfrac{\theta}{2\pi}\right)\pi r^2 \\ =\left(\dfrac{\theta}{2\cancel{\pi}}\right)\cancel{\pi} r^2 \\ =\dfrac{1}{2}r^2 \theta$

When we compute areas, we take increasingly smaller regions and sum up as an integral. The green section only depends on $\theta$$\theta$, so if we partitioned the green sector into many smaller sectors, then each sector would have a small angle $d\theta$$d\theta$.

So, the area of the polar curves is $A={\displaystyle \frac{1}{2}}{\int }_a^b{r}^{2}d\theta$$A=\dfrac{1}{2}\int_a^b{r^2d\theta}$

NOTE: Due to the $r^2$$r^2$ in the formula for the area inside a polar curve, you will need to use the half-/double-angle formula for ${\sin }^2(\theta )$$\sin^2{(\theta)}$ and ${\cos }^2(\theta )$$\cos^2{(\theta)}$.

Sketching Area Bounded by Polar Curves

Investigation 01

Find the following areas:

1. The area enclosed by $r=2+3\cos (\theta )$$r=2+3\cos{(\theta)}$ for ${\displaystyle 0\le \theta \le \frac{\pi }{3}}$$\displaystyle 0\leq\theta\leq\frac{\pi}{3}$

2. The area enclosed by $r=1+\sin (\theta )$$r=1+\sin{(\theta)}$ for ${\displaystyle \frac{\pi }{4}\le \theta \le \frac{\pi }{3}}$$\displaystyle \frac{\pi}{4}\leq\theta\leq\frac{\pi}{3}$

3. The area enclosed by one petal of $r=\sin (2\theta )$$r=\sin{(2\theta)}$ HINT: Graph the curve in DESMOS to find the angles which start and end one petal; use Mathematica to compute the intersection(s) in terms of the independent variable, $\theta$$\theta$; or Find total area enclosed by curve and divide the total area by the number of petals.

Use Technology to Rewrite a Polar Equation

NOTE: Since the area formula for polar curves contains $r^2$$r^2$, you will likely encounter ${\cos }^2(\theta )$$\cos^2{(\theta)}$ or ${\sin }^2(\theta )$$\sin^2{(\theta)}$. On all Assessments, you may use technology to rewrite sine and cosine functions raised to even powers without showing any work.

Mathematica

​x
1
(* Example from Investigation 01-2 *)
2
(* Use the variable t instead of theta *)
3

4
(* Rewrite integrand using a trig identity *)
5
TrigReduce[(2 + 3 Cos[t])^2]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. TrigReduce[ ] has one argument:

   1. The function to be rewritten.

3. For help on using the TrigReduce[ ] function:

   1. In Mathematica, execute the code: $\text{?TrigReduce}$$\text{?TrigReduce}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the TrigReduce[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Use Technology to Verify Area Enclosed by a Polar Curve

NOTE: On all Assessments, you may use technology to verify your computed area. You must show all Calculus II related work.

Mathematica

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1
(* Example from Investigation 01-2 *)
2
(* Use the variable t instead of theta *)
3

4
(* Find the area enclosed by the curve *)
5
1/2*Integrate[(2+3Cos[t])^2, {t,Pi/4,Pi/3}]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Integrate[ ] has at least two arguments:

   1. The function in the integrand;

   2. $\{t,a,b\}$$\{t,a,b\}$ where $t$$t$ is the independent variable; $a$$a$ is the lower limit of integration; $b$$b$ is the upper limit of integration.

3. For help on using the Integrate[ ] function:

   1. In Mathematica, execute the code: $\text{?Integrate}$$\text{?Integrate}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Integrate[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Use Technology to Solve Polar Equations Exactly

NOTE: On all Assessments, you may use technology to solve equations without showing any work.

Mathematica

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1
(* Example from Investigation 01-3 *)
2
(* from the DESMOS graph, a petal starts and ends at the 2origin, r=0 *)
3
(* Use the variable t instead of theta *)
4

5
(* Restrict answers to [0,2pi) *)
6
Solve[Sin[t]==Cos[2t] && 0 <= t < 2 Pi, t, Reals]
7

8
(* If the answers from the above code are not easy to interpret *)
9

10
(* Solve using the simplified trig equation *)
11
Solve[Cos[t]==Cos[2t] && 0 <= t < 2 Pi, t, Reals]//N

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Mathematica requires two $==$$==$ for the equals sign in an equation.

3. Solve[ ] has at least two arguments:

   1. The equation to be solved;

      1. $\text{\&\&}$$\text{\&\&}$ is Mathematica code for “and,” i.e., both equation and condition must be true

   2. The variable, i.e., $t$$t$, to be solved for;

   3. The argument $\text{Reals}$$\text{Reals}$ to force Mathematica to return only real-valued results.

4. Mathematica may return complex solutions; thus, either

   1. Identify a complex solution (the result includes the imaginary unit, $i$$i$), or

   2. Use the argument Reals.

5. For help on using the Solve[ ] function:

   1. In Mathematica, execute the code: $\text{?Solve}$$\text{?Solve}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Solve[ ] function – you will be able to copy-paste code.

6. Remember, correct Mathematica code will be all black except for variables.

7. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Investigation 02

Find the following areas:

1. The area that lies inside $r=2\sin (\theta )$$r=2\sin{(\theta)}$ and outside ${\displaystyle r=1}$$\displaystyle r=1$ HINT: Graph the curves in DESMOS and subtract areas.

2. The area that lies inside both $r=\sin (2\theta )$$r=\sin{(2\theta)}$ and ${\displaystyle r=\cos (2\theta )}$$\displaystyle r=\cos{(2\theta)}$
   HINT: Graph the curves in DESMOS or use Mathematica to find the exact intersections of the polar functions.

Use Technology to Exaclty Solve for Intesection of Polar Equations

NOTE: On all Assessments, you may use technology to solve equations without showing any work.

Mathematica

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11
1
(* Investigation 02-2 *)
2
(* To solve for intesection(s) of r=sin(2t) and r=cos(2t) *)
3
(* Use the variable t instead of theta *)
4

5
(* Restrict answers to [0,2pi) *)
6
Solve[Sin[2t]==0 && 0 <= t < 2 Pi, t, Reals]
7

8
(* If the answers from the above code are not easy to interpret *)
9

10
(* Solve using the simplified trig equation *)
11
Solve[Sin[2t]==0 && 0 <= t < 2 Pi, t, Reals]//N

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. Mathematica requires two $==$$==$ for the equals sign in an equation.

3. Solve[ ] has at least two arguments:

   1. The equation to be solved;

      1. $\text{\&\&}$$\text{\&\&}$ is Mathematica code for “and,” i.e., both equation and condition must be true

   2. The variable, i.e., $t$$t$, to be solved for;

   3. The argument $\text{Reals}$$\text{Reals}$ to force Mathematica to return only real-valued results.

4. Mathematica may return complex solutions; thus, either

   1. Identify a complex solution (the result includes the imaginary unit, $i$$i$), or

   2. Use the argument Reals.

5. For help on using the Solve[ ] function:

   1. In Mathematica, execute the code: $\text{?Solve}$$\text{?Solve}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the Solve[ ] function – you will be able to copy-paste code.

6. Remember, correct Mathematica code will be all black except for variables.

7. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Homework

At this time, you should be able to complete the following assignments:

• Section 10.4: # 1, 2, 3, 5--9, 11, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Tuesday, 3 November 2020 7:04 EDT Last Modified: Sunday, 31 October 2021 - 09:20 (EDT)