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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Finding the Center of MassIn Exercises 37–40, find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.

y = x 3 , y = 0 , x = 2 , ρ = k x

To determine

To calculate: The center of mass of given lamina is bounded by y=x3, y=0 and x=2.

Explanation

Given:

The lamina is bounded by y=x3, y=0 and x=2.

The density is ρ(x,y)=kx.

Formula used:

x¯=MYm,y¯=MXm

MY=Qxρ(x,y)dA is the moment of mass with respect to Y axis.

MX=Qyρ(x,y)dA is the moment of mass with respect to X axis.

m=Qρ(x,y)dA is the mass of the lamina, where ρ(x,y) is the density of the solid.

Calculation:

Plot the graph of equations y=x3, y=0 and x=2.

y=0 represents x- axis, x=2 represents vertical line parallel to y-axis intersecting x- axis at x=2.

y=x3 represents simple cubic curve,

For mass of lamina,

m=Qρ(x,y)dA=08y132kxdxdy=(k2)08[x2]y132dy=(k2)08(4y23)dy

Solve further,

m=(k2)[4y3y535]08=(325)k

Now,

MY=Qxρ(x,y)dA<

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Chapter 14 Solutions

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