   Chapter 10.3, Problem 87E

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 = t , y = 4 − t

To determine

To calculate: The centroid of region bounded by the curves x=t and y=4t.

Explanation

Given:

The parametric functions are x=t and y=4t, where 0<t<4.

Formula used:

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

The centroid of the bounded area along the x-axis

x¯=Total moments in x directionTotal Area

The centroid of the bounded area along the y-axis.

y¯=Total moments in x directionTotal Area

Calculation:

The parametric function is:

x=t

The parametric function x=t is defined for t0. So, the domain of the function x=t is, t0.

Another parametric function is:

y=4t

Now at t=0,

0=4tt=4

In the closed region of x=t and y=4t the value of t varies from, 0t4.

The area of the bounded region is,

A=04ydx

The parametric function is,

x=t

Differentiate the function with respect to t,

dxdt=12tdx=dt2t

Substitute the values of y and dx.

A=04(4t)12tdt=1204(4t12t12)dt

Integrate the equation,

A=12[8t23tt]04=12[8423440+0]=12=163

The centroid of the bounded area along the x-axis

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