Q. 36 Areas of regions bounded by pola... [FREE SOLUTION]

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Chapter 9: Q. 36 (page 775)

Areas of regions bounded by polar functions: Find the areas of the following regions. The area between the inner and outer loops of $r = 1 鈭� \sqrt{2} \cos 胃$

Short Answer

Expert verified

The required area is $\frac{4 蟺}{3} - \frac{\sqrt{3}}{4} + \sqrt{6} units$

Step by step solution

Given information.

$r = 1 鈭� \sqrt{2} \cos 胃$

Area

The graph of the function $r = 1 鈭� \sqrt{2} \cos 胃$ is

Now, find the limits of integration.

$0 = 1 鈭� \sqrt{2} \cos 胃鈬� \cos 胃 = \frac{1}{\sqrt{2}} 鈬� 胃 = \cos^{- 1} (\frac{1}{\sqrt{2}}) 鈬� 胃 = \frac{蟺}{3}, \frac{5 蟺}{3}$

Now, the area between the inner and outer loops is,

$A = \frac{1}{2} {鈭�}_{伪}^{尾} r^{2} d 胃 A = \frac{1}{2} {鈭�}_{\frac{蟺}{3}}^{\frac{5 蟺}{3}} {(1 - \sqrt{2} \cos 胃)}^{2} d 胃 A = \frac{1}{2} {鈭�}_{\frac{蟺}{3}}^{\frac{5 蟺}{3}} (1 + 2 \cos^{2} 胃 - 2 \sqrt{2} \cos 胃) d 胃 A = \frac{1}{2} {鈭�}_{\frac{蟺}{3}}^{\frac{5 蟺}{3}} (1 + 1 + \cos 2 胃 - 2 \sqrt{2} \cos 胃) d 胃 A = \frac{1}{2} {鈭�}_{\frac{蟺}{3}}^{\frac{5 蟺}{3}} (2 + \cos 2 胃 - 2 \sqrt{2} \cos 胃) d 胃 A = \frac{1}{2} {(2 胃 + \frac{\sin 2 胃}{2} - 2 \sqrt{2} \sin 胃)}_{\frac{蟺}{3}}^{\frac{5 蟺}{3}} A = \frac{1}{2} \{2 (\frac{5 蟺}{3}) + \frac{\sin 2 (\frac{5 蟺}{3})}{2} - 2 \sqrt{2} \sin (\frac{5 蟺}{3}) - [2 (\frac{蟺}{3}) + \frac{\sin 2 (\frac{蟺}{3})}{2} - 2 \sqrt{2} \sin (\frac{蟺}{3})]\} A = \frac{4 蟺}{3} - \frac{\sqrt{3}}{8} + \frac{\sqrt{6}}{2} - \frac{\sqrt{3}}{8} + \frac{\sqrt{6}}{2} A = \frac{4 蟺}{3} - \frac{\sqrt{3}}{4} + \sqrt{6} units$

The required area is $A = \frac{4 蟺}{3} - \frac{\sqrt{3}}{4} + \sqrt{6} units$

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Areas of regions bounded by polar functions: Find the areas of the following regions. The area between the inner and outer loops of $r = 1 鈭� \sqrt{2} \cos 胃$

Short Answer

Step by step solution

Given information.

Area

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Areas of regions bounded by polar functions: Find the areas of the following regions. The area between the inner and outer loops of r=1鈭�2cos胃

Short Answer

Step by step solution

Given information.

Area

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Calculus

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Areas of regions bounded by polar functions: Find the areas of the following regions. The area between the inner and outer loops of $r = 1 鈭� \sqrt{2} \cos 胃$