Explanation Given: The given curve function is r = sin θ and r = cos θ Formula Used: If f is a continuous function, then the area bounded by a curve in polar form r = f ( θ ) and the rays θ = α and θ = β ( with α < β ) is equal to 1 2 ∫ α β r 2 d θ = 1 2 ∫ α β ( f ( θ ) ) 2 d θ Graphing: The graph is shown below: Calculation: We first find the value of θ at the point of intersection of the two circles, by solving the equation sin θ = cos θ So, sin θ = cos θ tan θ = 1 θ = π 4 The region is divided into two parts, one bounded by r = sin θ for 0 ≤ θ ≤ π 4 and the other bounded by r = cos θ for π 4 ≤ θ ≤ π 2 . First, we compute the area of the region A 1 . The region is traced as θ varies from 0 to π 4 . So, α = 0 and β = π 4 The curve function is given by r = sin θ Substitute the value of α , β and r in an equation 1 2 ∫ α β r 2 d θ , we get A 1 = 1 2 ∫ 0 π 4 ( sin θ ) 2 d θ = 1 2 ∫ 0 π 4 sin 2 θ d θ = 1 2 ∫ 0 π 4 1 2 ( 1 − cos ( 2 θ ) ) d θ = 1 4 ∫ 0 π 4 ( 1 − cos 2 θ ) d θ = 1 4 | θ − 1 2 sin 2 θ | 0 π 4 = 1 4 ( ( π 4 − 1 2 sin 2 × π 4 ) − ( 0 − 1 2 sin 2 × 0 ) ) = π 16 − 1 8 Now, we compute the area of the region A 2 ...

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ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Chapter 12.4, Problem 12E

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The area of the two intersecting curves of the circle $r = \sin θ$ and $r = \cos θ$ .

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Chapter 12.4, Problem 11E

Chapter 12.4, Problem 13E

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Chapter 12 Solutions

EBK CALCULUS

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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The area of the two intersecting curves of the circle r = sin θ and r = cos θ .

Solution Summary: The author explains how to determine the area of the two intersecting curves of a circle by solving the equation sin = cos.

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The area of the two intersecting curves of the circle $r = \sin θ$ and $r = \cos θ$ .

Want to see the full answer?

Chapter 12 Solutions

The area of the two intersecting curves of the circle r = sin θ and r = cos θ .

Solution Summary: The author explains how to determine the area of the two intersecting curves of a circle by solving the equation sin = cos.

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The area of the two intersecting curves of the circle r=sinθ and r=cosθ.

Want to see the full answer?

Chapter 12 Solutions

The area of the two intersecting curves of the circle $r = \sin θ$ and $r = \cos θ$ .