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Area of an ellipse Consider the polar equation of an
Chapter 8, Problem 48RE(choose chapter or problem)
Area of an ellipse Consider the polar equation of an ellipse \(r=e d /(1 \pm e \cos \theta)\), where 0 < e < 1. Evaluate an integral in polar coordinates to show that the area of the region enclosed by the ellipse is \(\pi a b\), where 2a and 2b are the lengths of the major and minor axes, respectively.
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QUESTION:
Area of an ellipse Consider the polar equation of an ellipse \(r=e d /(1 \pm e \cos \theta)\), where 0 < e < 1. Evaluate an integral in polar coordinates to show that the area of the region enclosed by the ellipse is \(\pi a b\), where 2a and 2b are the lengths of the major and minor axes, respectively.
ANSWER:Solution 48REStep 1:Consider the polar equation of an ellipse r = .Here , 0 < e < 1 and the curve defined over any interval in of length 2.So , varies from 0 to 2.Let 2a , 2b are lengths of the major and minor axes , respectively.The equation of an ellipse is + =1.Consider the parametric equations x = r cos(, y = r sin(.Substitute these parametric equations in the equation + =1. + =1 Therefore , the equation of the ellipse in the polar form is ; .