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Chapter 9: Q. 25 (page 756)

In Exercises 17-25 find a definite integral expression that represents the area of the given region in the polar plane, and then find the exact value of the expression.
The graph of the polar equation $r = \sec 胃 - 2 \cos 胃$ for $胃鈭� (- \frac{蟺}{2}, \frac{蟺}{2})$ is called a strophoid. Graph the strophoid and find the area bounded by the loop of the graph.

Short Answer

Expert verified

Ans: The area of the equation is $A = (2 - \frac{蟺}{2})$

Step by step solution

01

Step 1. Given information:

The polar equation $r = \sec 胃 - 2 \cos 胃$

02

Step 2. Implying formula:

Formula to find the area is $A = {鈭�}_{伪}^{尾} \frac{1}{2} (f (胃))^{2} d 胃$ or $A = {鈭�}_{伪}^{尾} \frac{1}{2} r^{2} d 胃$ .

By equating, $\sec 胃 - 2 \cos 胃 = 0$

Thus,

$\sec 胃 = 2 \cos 胃 \cos 胃 = \frac{1}{\sqrt{2}} \cos 胃 = \cos \frac{蟺}{4} 胃 = \frac{蟺}{4}$

The limits are from 0 to $\frac{蟺}{4}$ .

03

Step 3. Continue:

Then,

$A = 2 路 \frac{1}{2} {鈭�}_{0}^{\frac{蟺}{4}} (\sec 胃 - 2 \cos 胃)^{2} d 胃 A = {鈭�}_{0}^{\frac{蟺}{4}} (\sec^{2} 胃 + 4 \cos^{2} 胃 - 2 路 2 \sec 胃路 \cos 胃) d 胃 A = {鈭�}_{0}^{\frac{蟺}{4}} (\sec^{2} 胃 + 4 \cos^{2} 胃 - 4) d 胃$

04

Step 4. Simplification:

A = {鈭�}_{0}^{\frac{蟺}{4}} (\sec^{2} 胃 + 4 \frac{1 + \cos 2 胃}{2} - 4) d [since \cos^{2} 胃 = \frac{1 + \cos 2 胃}{2}] A = {鈭�}_{0}^{\frac{蟺}{4}} (\sec^{2} 胃 + 2 + 2 \cos 2 胃 - 4) d 胃 A = (\tan 胃 - 2 胃 + \sin 2 胃)_{0}^{\frac{蟺}{4}}

05

Step 5. Continue:

Thus,

$A = (\tan \frac{蟺}{4} - 2 路 \frac{蟺}{4} + \sin 2 路 \frac{蟺}{4}) A = (1 - \frac{蟺}{2} + 1) A = (2 - \frac{蟺}{2})$

Therefore, the area of the equation is $A = (2 - \frac{蟺}{2})$ .

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Most popular questions from this chapter

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

$胃 = 蟺$

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

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Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22鈥�31.

$directrix x = 3, focus (0, 1)$

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

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