   Chapter 10.3, Problem 88E

Chapter
Section
Textbook Problem

# Centroid In Exercises 87 and 88, find the centroid of the region bounded by the graph of the parametric equations and the coordinate axes. (Use the result of Exercise 77.) x = 4 − 1 , y = t

To determine

To calculate: Centroid of region bounded by curves x=4t and y=t.

Explanation

Given:

The parametric function is, x=4t and y=t where, 0<t<4.

Formula used:

The formula for the centroid is, x¯=1Aabxydx, where A is the area.

Centroid of the bounded region along the x-axis,

x¯=1Aabxf(x)dx

Centroid of the bounded region along the y-axis,

y¯=1Aabyf(x)dx

Calculation:

The parametric function is, 4t0 and t0

Simplify the inequality,

0t4

Hence, the function is defined for 0t4.

Find the value of area with the help of the formula:

A=04ydx

The parametric function is, y=t.

Differentiate the parametric function:

dxdt=124tdx=dt24t

Substitute the values of y and dx,

A=0412(t4t)dt …… (1)

Now, u=4t

Differentiate the equation with respect to t.

dudt=124tdu=dt24tdt=2udu

And change the limits if

At t=4,

u=0

At t=0,

u=2

The region bounded is obtained from equation (1),

A=0212(4u2u)(2udu)=024u2du=0222u2du=[u4u22+42sin1(u2)]02

On simplifying the equation,

A={(2)4(2)22+42sin1(22)}{(0)4(0)22+

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