   Chapter 8.4, Problem 55E

Chapter
Section
Textbook Problem

# Area Find the Area enclosed by the ellipse x 2 a 2 + y 2 b 2 = 1 . shown in the figure. To determine
Area enclosed by the ellipse x2a2+y2b2=1.

Explanation

Given: The ellipse x2a2+y2b2=1:

Refer to the question for figure.

Solution: The area of the given ellipse is πab._

Explanation: x2a2+y2b2=1

y2=b2(1x2a2)y=baa2x2

Note that the ellipse is symmetrical about the x axis.

Therefore A=2aabaa2x2dx

Also note that ellipse is symmetrical about the y axis.

A=4aabaa2x2dx

i.e; Area is 4 times the area of the first quadrant of the ellipse.

Because a2x2 is a standard form

We use the trigonometric substitution:

x=asinθdx=acosθdθx=0sinθ=0θ=0x=asin

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