Area Arc Length And Surface Area In Polar Equations Homework

We can use the equation of a curve in polar coordinates to compute some areas bounded by such curves. The basic approach is the same as with any application of integration: find an approximation that approaches the true value. For areas in rectangular coordinates, we approximated the region using rectangles; in polar coordinates, we use sectors of circles, as depicted in figure 10.3.1. Recall that the area of a sector of a circle is $\ds \alpha r^2/2$, where $\alpha$ is the angle subtended by the sector. If the curve is given by $r=f(\theta)$, and the angle subtended by a small sector is $\Delta\theta$, the area is $\ds (\Delta\theta)(f(\theta))^2/2$. Thus we approximate the total area as $$\sum_{i=0}^{n-1} {1\over 2} f(\theta_i)^2\;\Delta\theta.$$ In the limit this becomes $$\int_a^b {1\over 2} f(\theta)^2\;d\theta.$$

Example 10.3.1 We find the area inside the cardioid $r=1+\cos\theta$. $$\int_0^{2\pi}{1\over 2} (1+\cos\theta)^2\;d\theta= {1\over 2}\int_0^{2\pi} 1+2\cos\theta+\cos^2\theta\;d\theta= {1\over 2}\left.(\theta +2\sin\theta+ {\theta\over2}+{\sin2\theta\over4})\right|_0^{2\pi}={3\pi\over2}.$$

Figure 10.3.1. Approximating area by sectors of circles.

Example 10.3.2 We find the area between the circles $r=2$ and $r=4\sin\theta$, as shown in figure 10.3.2. The two curves intersect where $2=4\sin\theta$, or $\sin\theta=1/2$, so $\theta=\pi/6$ or $5\pi/6$. The area we want is then $$ {1\over2}\int_{\pi/6}^{5\pi/6} 16\sin^2\theta-4\;d\theta={4\over3}\pi + 2\sqrt{3}. $$

Figure 10.3.2. An area between curves.

This example makes the process appear more straightforward than it is. Because points have many different representations in polar coordinates, it is not always so easy to identify points of intersection.

Example 10.3.3 We find the shaded area in the first graph of figure 10.3.3 as the difference of the other two shaded areas. The cardioid is $r=1+\sin\theta$ and the circle is $r=3\sin\theta$. We attempt to find points of intersection: $$\eqalign{ 1+\sin\theta&=3\sin\theta\cr 1&=2\sin\theta\cr 1/2&=\sin\theta.\cr} $$ This has solutions $\theta=\pi/6$ and $5\pi/6$; $\pi/6$ corresponds to the intersection in the first quadrant that we need. Note that no solution of this equation corresponds to the intersection point at the origin, but fortunately that one is obvious. The cardioid goes through the origin when $\theta=-\pi/2$; the circle goes through the origin at multiples of $\pi$, starting with $0$.

Now the larger region has area $$ {1\over2}\int_{-\pi/2}^{\pi/6} (1+\sin\theta)^2\;d\theta= {\pi\over2}-{9\over16}\sqrt{3} $$ and the smaller has area $$ {1\over2}\int_{0}^{\pi/6} (3\sin\theta)^2\;d\theta= {3\pi\over8} - {9\over16}\sqrt{3} $$ so the area we seek is $\pi/8$.

Figure 10.3.3. An area between curves.

Exercises 10.3

Find the area enclosed by the curve.

Ex 10.3.1 $\ds r=\sqrt{\sin\theta}$ (answer)

Ex 10.3.2 $\ds r=2+\cos\theta$ (answer)

Ex 10.3.3 $\ds r=\sec\theta, \pi/6\le\theta\le\pi/3$ (answer)

Ex 10.3.4 $\ds r=\cos\theta, 0\le\theta\le\pi/3$ (answer)

Ex 10.3.5 $\ds r=2a\cos\theta, a>0$ (answer)

Ex 10.3.6 $\ds r=4+3\sin\theta$ (answer)

Ex 10.3.7 Find the area inside the loop formed by $\ds r=\tan(\theta/2)$. (answer)

Ex 10.3.8 Find the area inside one loop of $\ds r=\cos(3\theta)$. (answer)

Ex 10.3.9 Find the area inside one loop of $\ds r=\sin^2\theta$. (answer)

Ex 10.3.10 Find the area inside the small loop of $\ds r=(1/2)+\cos\theta$. (answer)

Ex 10.3.11 Find the area inside $\ds r=(1/2)+\cos\theta$, including the area inside the small loop. (answer)

Ex 10.3.12 Find the area inside one loop of $\ds r^2=\cos(2\theta)$. (answer)

Ex 10.3.13 Find the area enclosed by $r=\tan\theta$ and $\ds r={\csc\theta\over\sqrt2}$. (answer)

Ex 10.3.14 Find the area inside $r=2\cos\theta$ and outside $r=1$. (answer)

Ex 10.3.15 Find the area inside $r=2\sin\theta$ and above the line $r=(3/2)\csc\theta$. (answer)

Ex 10.3.16 Find the area inside $r=\theta$, $0\le\theta\le2\pi$. (answer)

Ex 10.3.17 Find the area inside $\ds r=\sqrt{\theta}$, $0\le\theta\le2\pi$. (answer)

Ex 10.3.18 Find the area inside both $\ds r=\sqrt3\cos\theta$ and $r=\sin\theta$. (answer)

Ex 10.3.19 Find the area inside both $r=1-\cos\theta$ and $r=\cos\theta$. (answer)

Ex 10.3.20 The center of a circle of radius 1 is on the circumference of a circle of radius 2. Find the area of the region inside both circles. (answer)

Ex 10.3.21 Find the shaded area in figure 10.3.4. The curve is $r=\theta$, $0\le\theta\le3\pi$. (answer)

Figure 10.3.4. An area bounded by the spiral of Archimedes.

Chapter 10:  Parametric Equations and Polar Coordinates

0) Textbook Problems:  Ch. 10.2      Ch. 10.3      Ch. 10.4      Ch. 10.5         Ch.10 Review Problems                   Ch. 10 Odd Problem Answers

1) 10.2 Parametric Equations:   notes    notes.video      Homework

                                                      10.2 BC HW 2017

2) 10.3a Parametric Equations &Derivatives:  notes      notes1.video       notes2.video     WS    Solutions                                                                               10.3a BC HW 2017


3)10.3b Parametric Equations/Calculus Vectors: notes   notes1.video  notes2.video      WS      Solutions                                                                                    10.3b BC HW 2017                           Homework         WS #2 


4) 10.4a Graphing Polar Equations: notes     notes1.video       notes2.video  
                                                             
10.4a BC HW 2017

5) 10.4b Polar Equations and Derivatives:notes    notes1.video        notes2.video       WS            Solutions                                                                       10.4b  BC HW 2017                                     10.4 Homework  

​6) 10.5 Polar Area and Arc Length:  notes    notes1.video        notes2.video           WS             Solutions   
                                                             10.5 BC HW 2017                 10.5 Homework 1      10.5 Homework 2

7) Ch. 10 Additional Notes Area Surface of Revolution (Parametric and Polar Form): notes 

8)10.2 - 10.5 Review: 
              10.2-10.5 Review Worksheet #1          Review1.video1       Review1.video2         Review1.video3 

              10.2-10.5 Review Worksheet #2          Review2.video1       Review2.video2        

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