   Chapter 10.2, Problem 31E

Chapter
Section
Textbook Problem

Use the parametric equations of an ellipse, x = a cos θ, y = b sin θ, 0 ≤ θ ≤ 2π, to find the area that it encloses.

To determine

To find: The area that ellipse encloses for the parametric equations of ellipse x=acosθ and y=bsinθ .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=acosθ

The parametric equation for the variable y is as follows.

y=bsinθ

The range of values of angle θ is 0 to 2π .

Calculation:

Use the rule of substitution for definite integrals to calculate area.

A=βag(t)f'(t)dt

Differentiate the parametric equation x with respect to θ .

x=acosθdxdθ=asinθdx=asinθdθ

The ellipse is symmetric about the center about the x axis and y axis.

The expression for area is A=θ1θ2ydx .

Substitute π2 for θ1 and 0 for θ2 .

A=4π20ydx

Substitute asinθdθ for dx

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