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Show that when the density of the region is constant, the first moment about the x-axis is

Mx=12-x+22x-1ydydx=2

Short Answer

Expert verified

The first moment of the mass in Ωabout the xaxis isMx=2

Step by step solution

01

Given information

The density of the region is constant, the first moment about the x-axis is

Mx=12-x+22x-1ydydx=2

02

Step 2:Simplification

The objective of this problem is to show that when the density of region is constant the first moment about the xaxis is

Mx=12-x+22x-1ydydx=2

Plot the vertices localid="1650646891787" (1,1),(2,0), and localid="1650646895001" (2,3)and join them.

First moment of the mass in Ωabout the xaxis is

Mx=yρ(x,y)dA

Where ρ(x,y)is the density of the region Ω.

Here ρ(x,y)is constant.

Assume ρ(x,y)=k. Then

Mx=ykdA

Impose the limits on integrals.

Mx=12-x+22x-1kydydx

Integrate the inner integral first

Mx=k12-x+22x-1ydydx

Integrate with respect to y

Mx=12y22-x+22x-1dx[Takek=1]

Substitute the limits

Mx=12(2x-1)22-(-x+1)22dxMx=124x2-4x+12-x2-2x+12dxMx=12123x2-2xdx[Simplify]

Integrate with respect to x

Mx=1233x3-22x212

Substitute the limits

Mx=(2)3-(2)2-(1)3+(1)2Mx=12[4]Mx=2[Simplify]

Thus, the first moment of the mass in Ωabout the xaxis is

Mx=2

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