Explanation Given information: The ellipse ( x a ) 2 + ( y b ) 2 = 1 parametrize the boundary by: x = a cos θ , y = b sin θ where 0 ≤ θ ≤ 2 π Concept used: Area of the ellipse by Green’s Theorem: Enclosed Area = ∮ c x d y = ∮ c − y d x Enclosed Area ⇒ 1 2 ∮ c x d y − y d x Identity of cos ( 2 θ ) = 2 cos 2 θ − 1 cos 2 θ = 1 2 ( 1 + cos ( 2 θ ) ) Identity of sin 2 θ = 1 2 [ 1 − cos ( 2 θ ) ] Calculation: First differentiate x = a cos θ , y = b sin θ with respect to x and y respectively: x = a cos θ d d x ( x ) = d d x ( a cos θ ) d x = − sin θ d θ y = a sin θ d d x ( y ) = d d x ( a sin θ ) d y = a cos θ d θ Now, calculate the enclosed area: Enclosed area = ∮ c x d y Put the value of x and d y in above equation Enclosed area = ∮ c x d y = ∫ 0 2 π ( a cos θ ) . ( b cos θ ) d θ = ∫ 0 2 π a b cos 2 θ d θ = a b ∫ 0 2 π cos 2 θ d θ = a b ∫ 0 2 π [ 1 2 ( 1 + cos ( 2 θ ) ) ] d θ = a b ∫ 0 2 π [ 1 2 d θ + 1 2 cos ( 2 θ ) d θ ] = a b [ θ 2 + 1 4 sin ( 2 θ ) ] 0 2 π = a b [ 2 π 2 + 1 4 sin [ 2 × 2 π ] − ( 0 2 + 1 4 sin [ 2 × 0 ] ) ] = a b [ π + 1 4 sin [ 4 π<

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Chapter 17.1, Problem 17E

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Compute the area of the ellipse (xa)2+(yb)2=1 using 12∮cxdy−ydx.

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Chapter 17.1, Problem 16E

Chapter 17.1, Problem 18E

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Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

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Compute the area of the ellipse ( x a ) 2 + ( y b ) 2 = 1 using 1 2 ∮ c x d y − y d x .

Solution Summary: The author calculates the area of the ellipse by Green’s Theorem: lEnclosed Area=displaystyle

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Compute the area of the ellipse (xa)2+(yb)2=1 using 12∮cxdy−ydx.

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Chapter 17 Solutions