Chapter 8, Problem 77RE

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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=1âˆ’x2 where y=0.

Formula used:

Formula for centroid,

yÂ¯=MXA

Where, MX=âˆ«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=1âˆ’x2.

Table for different values of x and y:

xy=1âˆ’x2âˆ’10010.50.86610

The graph obtained is:

The graph obtained is a semicircle of radius 1.

Area of the semi-circular region,

A=Ï€(1)22=Ï€2

By symmetry,

xÂ¯=0

Now,

MX=âˆ«RydA=âˆ«01y(2xdy)=2âˆ«01y1âˆ’y2dy

Let 1âˆ’y2=u

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