   Chapter 15.4, Problem 4E

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.4. D = {(x, y) | 0 ≤ x ≤ a, 0 ≤ y ≤ b}; ρ(x, y) = 1 + x2 + y2

To determine

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

Explanation

Given:

The region D is D={(x,y)|0xa,0yb}.

The density function is ρ(x,y)=1+x2+y2.

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

Calculation:

The total mass of the lamina is,

m=Dρ(x,y)dA=0a0b(1+x2+y2)dydx.

Integrate with respect to y and apply it’s limit.

m=0a(y+x2y+y33)0bdx=0a[(b+x2b+b33)(0+x2(0)+033)]dx=0a[(b+bx2+b33)(0+0+0)]dx=0a(b+bx2+b33)dx

Integrate with respect to x and apply it’s limit.

m=[bx+bx33+b3x3]0a=(ba+ba33+b3a3)(b(0)+b(0)33+b3(0)3)=(ab+a3b3+ab33)(0+0+0)=ab3(3+a2+b2)

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

x¯=1mDxρ(x,y)dA=1ab3(3+a2+b2)0a0bx(1+x2+y2)dydx=3ab(3+a2+b2)0a0b(x+x3+xy2)dydx

Integrate with respect to y and apply the corresponding limit.

x¯=3ab(3+a2+b2)0a(xy+x3y+xy33)0bdx=3ab(3+a2+b2)0a[(xb+x3b+xb33)(x(0)+x3(0)+x(0)33)]dx=3ab(3+a2+b2)0a[(bx+bx3+xb33)(0+0+0)]dx=3ab(3+a2+b2)0a(bx+bx3+xb33)dx

Integrate with respect to x and apply it’s limit

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