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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
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[16163]=163

The centroid of the bounded area along the x-axis

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