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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A lamina occupies the part of the disk x2 + y2a2 that lies in the first quadrant.

(a) Find the centroid of the lamina.

(b) Find the center of mass of the lamina if the density function is ρ(x, y) = xy2.

(a)

To determine

To find: The centroid of the lamina occupied by the given disk D.

Explanation

Given:

The region D is the disk x2+y2a2 in the first quadrant.

The density function is proportional to x-axis, that is ρ(x,y)=xy2.

Formula used:

The centroid of the lamina that occupies the given region D is (x¯,y¯).

Here, x¯=1ADxdA and y¯=1ADydA where, A is the area of the given region.

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π, then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

Here, the region is the circle of radius a. So, use of polar coordinates is a wise choice. Therefore, from the given conditions it is observed that r varies from 0 to a and θ varies from 0 to π2. The area of the region is given by,

A=πr24=πa24

Then, by the equation (1) and (2), the centroid of the lamina is,

x¯=1(πa24)DxdA=4πa20a0π2rcosθ(r)dθdr=4πa20π2cosθdθ0ar2dr=</

(b)

To determine

To find: The center of mass of the lamina occupied by the given disk D.

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