   Chapter 10.5, Problem 63E

Chapter
Section
Textbook Problem

# Find the centroid of the region enclosed by the x-axis and the top half of the ellipse 9x2 + 4y2 = 36.

To determine

To find: The centroid of the region enclosed by x axis and the top half of the ellipse is 9x2+4y2=36 .

Explanation

Given:

The equation of the ellipse is 9x2+4y2=36 .

Calculation:

Compute the value of a and b .

9x2+4y2=36 (1)

Divide the above equation by the value 36 on both the sides.

9x236+4y236=3636x24+y29=1

Compare the above equation with standard equation of the ellipse

x2b2+y2a2=1x24+y29=1

Then,

a2=9a=9a=3

b2=4b=4b=2

Rewrite the equation (1) to get y :

9x2+4y2=364y2=369x2y2=94(4x2)y=94(4x2)y=324x2y=±324x2f(x)=±324x2

The formula for area of the ellipse is as below:

A=πab

For the half ellipse. the formula of the ellipse is as below.

A=12πab

Substitute the value of 3 for a and 2 for b .

A=12×π×3×2A=3π

Compute the centroid of the region enclosed by x axis as below

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