Chapter 9: Q. 49. (page 756)

Find the area interior to two circles with the same radius
if each circle passes through the center of the other. (Hint:
Consider the circles $r = a a n d r = r = 2 a \cos θ$

Short Answer

Expert verified

Therefore the required area is $a^{2} (\frac{2 π}{3} - \frac{\sqrt{3}}{2})$

Step by step solution

Given information

Take a look at the polar function

$r = a, r = 2 a \cos θ$

The objective is to find the area interior of two circles with the same radius.

To find the limits, equal the functions

$a = 2 a \cos θ \Rightarrow \frac{a}{2 a} = \cos θ \Rightarrow \cos θ = \frac{1}{2} \Rightarrow θ = \frac{π}{3}, \frac{2 π}{3}$

Find the area interior of the circles

The region's corresponding limits are $0$ to $\frac{2 π}{3}$

The interval is $[0, \frac{2 π}{3}]$

Formula to find the area is $A = \int_{α}^{β} \frac{1}{2} (f (θ))^{2} d θ$ or $A = \int_{α}^{β} \frac{1}{2} r^{2} d θ$

The area between the circles,

role="math" localid="1653848508245" $A = 2 \cdot \frac{1}{2} \int_{0}^{\frac{2 π}{3}} ((2 a \cos θ)^{2} - a^{2}) d θ \Rightarrow A = \int_{0}^{\frac{2 π}{3}} (4 a^{2} \cos^{2} θ - a^{2}) d θ \Rightarrow A = a^{2} \int_{0}^{\frac{2 π}{3}} (4 \cos^{2} θ - 1) d θ \Rightarrow A = a^{2} \int_{0}^{\frac{2 π}{3}} (4 \frac{1 + \cos 2 θ}{2} - 1) d θ [S i n c e \cos 2 θ = 2 \cos^{2} θ - 1 \Rightarrow \cos^{2} θ = \frac{1 + \cos 2 θ}{2}] \Rightarrow A = a^{2} \int_{0}^{\frac{2 π}{3}} (2 (1 + \cos 2 θ) - 1) d θ \Rightarrow A = a^{2} \int_{0}^{\frac{2 π}{3}} (2 + 2 \cos 2 θ - 1) d θ$

On integration,

$A = a^{2} {(θ + 2 \frac{\sin 2 θ}{2})}_{0}^{\frac{2 π}{3}}$

Find the area by applying the limits

By applying the limits,
$A = a^{2} (\frac{2 π}{3} + \sin 2 \cdot \frac{2 π}{3} - 0) \Rightarrow A = a^{2} (\frac{2 π}{3} - \frac{\sqrt{3}}{2}) [since \sin \frac{4 π}{3} = - \frac{\sqrt{3}}{2}]$

Hence, the required area is $a^{2} (\frac{2 π}{3} - \frac{\sqrt{3}}{2})$

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Find the area interior to two circles with the same radius
if each circle passes through the center of the other. (Hint:
Consider the circles $r = a a n d r = r = 2 a \cos θ$

Short Answer

Step by step solution

Given information

The objective is to find the area interior of two circles with the same radius.

Find the area interior of the circles

Find the area by applying the limits

One App. One Place for Learning.

Most popular questions from this chapter

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Find the area interior to two circles with the same radiusif each circle passes through the center of the other. (Hint:Consider the circles r=aandr=r=2acosθ

Short Answer

Step by step solution

Given information

The objective is to find the area interior of two circles with the same radius.

Find the area interior of the circles

Find the area by applying the limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Calculus

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find the area interior to two circles with the same radius
if each circle passes through the center of the other. (Hint:
Consider the circles $r = a a n d r = r = 2 a \cos θ$