   Chapter 14.4, Problem 19E

Chapter
Section
Textbook Problem

Finding the Center of MassIn Exercises 13–24, find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. y = 4 − x 2 , y = 0 , ρ = k y

To determine

To calculate: The mass and center of mass of the lamina bounded by the graphs of the given equations.

Explanation

Given:

y=4x2,y=0,ρ=ky

Calculation:

First, sketch the graph as shown below,

Draw the reference axis in xy coordinates.

Then, plot the provided equations:

y=4x2,y=0.

 X -2 -1 0 1 2 Y 0 3 4 3 4

Further, mark the points on the plot and connect them.

Shade the region inside the graph as the area of interest is the area inscribed by the equations.

The final plot is shown below as,

Then, the mass of the lamina is calculated as follows,

m=2204x2kydydx=22[ky22]04x2dx=22k2(168x2+x4)dx=[k2(16x8x33+x55)]22m=25615k

Now, to find the center of the mass calculate the moment of inertia about both the axes to find the center of the mass.

The moment of inertia about x-axis is,

Mx=2204x2ky2dydxMx=22[ky33]04x2dxMx=22

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