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 y d x 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 y d x Put the value of y and d x in above equation Enclosed area = ∮ c y d x = ∫ 0 2 π y d x = ∫ 0 2 π ( b sin θ )

4th Edition

ISBN: 9781319279813

Author: Rogawski

Publisher: MAC HIGHER

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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 16E

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Compute the area of the ellipse (xa)2+(yb)2=1 using ∮cydx.

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Chapter 17.1, Problem 15E

Chapter 17.1, Problem 17E

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Chapter 17 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

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

Solution Summary: The author calculates the area of the ellipse (xa)2+

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Compute the area of the ellipse (xa)2+(yb)2=1 using ∮cydx.

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Chapter 17 Solutions