   Chapter 8, Problem 77RE

Chapter
Section
Textbook Problem

Centroid In Exercises 77 and 78, find the centroid of the region bounded by the graphs of the equations using any method. y = 1 − x 2 ,   y = 0

To determine

To calculate: The centroid of region that is bounded by graphs of the equations y=1x2 and y=0.

Explanation

Given:

The region is bounded by y=1x2 where y=0.

Formula used:

Formula for centroid,

y¯=MXA

Where, MX=RydA is the first moment of area of region R about X axis.

Here, A is the area of the region.

Calculation:

To draw the graph of the region, Find the corresponding values of the y for the different values of x by using the equation y=1x2.

Table for different values of x and y:

xy=1x210010.50.86610

The graph obtained is:

The graph obtained is a semicircle of radius 1.

Area of the semi-circular region,

A=π(1)22=π2

By symmetry,

x¯=0

Now,

MX=RydA=01y(2xdy)=201y1y2dy

Let 1y2=u

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