Explanation The given equation of the ellipse is r = e d 1 ± e cos θ note that the given equation contains the term cos θ so the directrices of the ellipse are vertical. Thus the corresponding equation in the Cartesian form will be x 2 a 2 + y 2 b 2 = 1 . To calculate the area convert this equation into polar form by substituting x = r cos θ , y = r sin θ in x 2 a 2 + y 2 b 2 = 1 . ( r cos θ ) 2 a 2 + ( r sin θ ) 2 b 2 = 1 r 2 b 2 cos 2 θ + r 2 a 2 sin 2 θ = a 2 b 2 r 2 = a 2 b 2 a 2 sin 2 θ + b 2 cos 2 r = a b a 2 sin 2 θ + b 2 cos 2 Use this equation to calculate the area. The ellipse is symmetric about the x -axis as well as the y -axis. The area of the total ellipse is 4 times the area of the region in the first quadrant. Note that the area in the first quadrant is generated when θ varies from 0 to π 2 . Thus, total area of the ellipse is calculated as follows. A = 4 [ 1 2 ∫ 0 π 2 ( a b a 2 sin 2 θ + b 2 cos 2 θ ) 2 d θ ] = 2 [ ∫ 0

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Question

Chapter 10, Problem 64RE

To determine

To show: That the area of the region inside an ellipse r=ed1±ecosθ is πab where the length of major axis is 2a and length of minor axis is 2b.

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Chapter 10, Problem 63RE

Chapter 10, Problem 65RE

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Chapter 10 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 10.1 - Explain how a pair of parametric equations...Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Give parametric equations that generate the line...Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Explain how to find points on the curve x = f(t),...

Knowledge Booster

That the area of the region inside an ellipse r = e d 1 ± e cos θ is π a b where the length of major axis is 2 a and length of minor axis is 2 b .

Solution Summary: The author explains that the given equation of the ellipse is ed1pm emathrmcostheta.

To show: That the area of the region inside an ellipse r=ed1±ecosθ is πab where the length of major axis is 2a and length of minor axis is 2b.

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Chapter 10 Solutions