Let A=(0,2),B=(1,0),O=(0,0).
Line OA: x=0,0≤y≤2
LineAB: y=−2x+2,0≤x≤1
Line OB: y=0,0≤x≤1
Given ρ(x,y)=1+3x+3y
Find the mass of the lamina
m=∬Dρ(x,y)dA=∫01∫02−2x(1+3x+3y)dydx
=∫01[y+3xy+23y2]2−2x0dx
=∫01(2−2x+6x−6x2+6−12x+6x2)dx
=[−4x2+8x]10=4(units of mass)Mass of the lamina is 4 units of mas.
Find the coordinates of the center of mass
xˉ=m1∬Dxδ(x,y)dA
=41∫01∫01−xx(1+3x+3y)dydx
=41∫01x[y+3xy+23y2]2−2x0dx
=41∫01(2x−2x2+6x2−6x3)dx
+41∫01(6x−12x2+6x3)dx
=41[−38x3+4x2]10=31
yˉ=m1∬Dyρ(x,y)dA
=41∫01∫01−xy(1+3x+3y)dydx
=41∫01[2y2+23xy2+y3]2−2x0dx
=41∫01(2−4x+2x2+6x−12x2+6x3)dx
+41∫01(8−24x+24x2−8x3)dx
=41[−21x4+314x3−11x2+10x]10=2419
Center of mass (31,2419)
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