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 $r = 1 - \sqrt{2} \cos θ$

Short Answer

Expert verified

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

Step by step solution

Given information.

$r = 1 - \sqrt{2} \cos θ$

Area

The graph of the function $r = 1 - \sqrt{2} \cos θ$ is

Now, find the limits of integration.

$0 = 1 - \sqrt{2} \cos θ \Rightarrow \cos θ = \frac{1}{\sqrt{2}} \Rightarrow θ = \cos^{- 1} (\frac{1}{\sqrt{2}}) \Rightarrow θ = \frac{π}{3}, \frac{5 π}{3}$

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

$A = \frac{1}{2} \int_{α}^{β} r^{2} d θ A = \frac{1}{2} \int_{\frac{π}{3}}^{\frac{5 π}{3}} {(1 - \sqrt{2} \cos θ)}^{2} d θ A = \frac{1}{2} \int_{\frac{π}{3}}^{\frac{5 π}{3}} (1 + 2 \cos^{2} θ - 2 \sqrt{2} \cos θ) d θ A = \frac{1}{2} \int_{\frac{π}{3}}^{\frac{5 π}{3}} (1 + 1 + \cos 2 θ - 2 \sqrt{2} \cos θ) d θ A = \frac{1}{2} \int_{\frac{π}{3}}^{\frac{5 π}{3}} (2 + \cos 2 θ - 2 \sqrt{2} \cos θ) d θ A = \frac{1}{2} {(2 θ + \frac{\sin 2 θ}{2} - 2 \sqrt{2} \sin θ)}_{\frac{π}{3}}^{\frac{5 π}{3}} A = \frac{1}{2} \{2 (\frac{5 π}{3}) + \frac{\sin 2 (\frac{5 π}{3})}{2} - 2 \sqrt{2} \sin (\frac{5 π}{3}) - [2 (\frac{π}{3}) + \frac{\sin 2 (\frac{π}{3})}{2} - 2 \sqrt{2} \sin (\frac{π}{3})]\} A = \frac{4 π}{3} - \frac{\sqrt{3}}{8} + \frac{\sqrt{6}}{2} - \frac{\sqrt{3}}{8} + \frac{\sqrt{6}}{2} A = \frac{4 π}{3} - \frac{\sqrt{3}}{4} + \sqrt{6} units$

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

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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 $r = 1 - \sqrt{2} \cos θ$

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=1−2cosθ

Short Answer

Step by step solution

Given information.

Area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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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 $r = 1 - \sqrt{2} \cos θ$