   Chapter 15.4, Problem 3E

Chapter
Section
Textbook Problem

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.3. D = {(x, y) | 1 ≤ x ≤ 3, 1 ≤ y ≤ 4}; ρ(x, y) = ky2

To determine

To find: The total mass and the center of mass of the lamina.

Explanation

Given:

The region D is D={(x,y)|1x3,1y4}.

The density function is ρ(x,y)=ky2.

Formula used:

The total mass of the lamina is, m=limk,li=1kj=1lρ(xij*,yij*)ΔA=Dρ(x,y)dA.

Here, the density function is given by ρ(x,y) and D  is the region that is occupied by the lamina.

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

Here, x¯=Mym=1mDxρ(x,y)dA and y¯=Mxm=1mDyρ(x,y)dA

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 (1)

Calculation:

The total mass of the lamina is,

m=Dρ(x,y)dA=1314ky2dydx

Integrate with respect to x and y by using (1),

m=k13dx14y2dy=k[y33]14[x]13

Apply the limit value of x and y,

m=k(433133)(31)=k(64313)(2)=k(633)(2)=2k(21)

=42k

In order to get the coordinates of the center of mass, find x¯ and y¯

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