   Chapter 10.2, Problem 53E

Chapter
Section
Textbook Problem

Show that the total length of the ellipse x = a sin θ, y = b cos θ, a > b > 0, is L = 4 a ∫ 0 π / 2 1 − e 2 sin 2 θ   d θ where e is the eccentricity of the ellipse (e = c/a, where c = a 2 − b 2 ).

To determine

To find: The total length of the ellipse for the parametric equation x=asinθ and y=bcosθ.

Explanation

Given:

The parametric equation for the variable x is as below.

x=asinθ

The parametric equation for the variable y is as below.

y=bcosθ

Where t ranges from 0 to π2.

Calculation:

The total length of the ellipse is obtained by, L=4×0π2(dxdθ)2+(dydθ)2dθ

Differentiate the variable x with respect to t.

dxdθ=asinθ=acosθ

Differentiate the variable y with respect to t:

dydθ=bcosθ=bsinθ

Ellipse is divided into four equal parts, and in first quadrant the value of θ varies from 0 to π2.

Write the total length of the ellipse formula for 4 equal parts.

L=4×0π2(dxdθ)2+(dydθ)2dθ

Substitute (acosθ) for dxdt and (bsinθ) for dydt in the above equation.

L=4×0π2(dxdθ)2+(dydθ)2dθ=0π2(acosθ)2+(bsinθ)2dθ=4×

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