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If the density at each point in T is proportional to the point's distance from the x-axis, find the mass of T.

Short Answer

Expert verified

The centroid of the triangular region is

x¯=34,y¯=0

Step by step solution

01

Given information

Vertices of the triangular region are (0,0),(1,1),and(1,-1).

02

calculation

The objective of this problem is to find the center of mass of the triangular region.

The density at each point is proportional to the point's distance from the y- axis. Density ρ(x,y)=kx

Use formula for center of mass

localid="1650641181816" x¯=Ωxρ(x,y)dAΩρ(x,y)dA and localid="1650641192893" y¯=Ωyρ(x,y)dAΩρ(x,y)dA

Useρ(x,y)=kx

localid="1650641380814" x¯=01-xxxkxdydx01-xxkxdydxx¯=01-xxkx2dydx01-xxkxdydxx¯=01kx2[y]-xxdx01kx[y]-xxdxx¯=01kx2[2x]dx01kx[2x]dxx¯=01kx3dx01kx2dxx¯=x441kx3301x¯=34

Now

y¯=Ωyρ(x,y)dA0ρ(x,y)dAy¯=01-xxykxdydx01-xxkxdydxy¯=01kxy22-xxdx01kx[y]-xxdxy¯=01kx[0]dx012kx2y¯=01kx[0]dxk23x301y¯=023k=0y¯=0

Thus, the centroid of the triangular region is

x¯=34,y¯=0

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