Given, the region D is bounded by the parabola x = 1 - y2 and the coordinate axes in the first quadrant with density function p (x, y) = y.
Moments,
Mxβ=β«β«yp(x,y)dA=β«x=01ββ«y=01βyp(x,y)dA=β«x=01ββ«y=01βy2dydx=β«x=01β[3y3β]01βdydx=31ββ«x=01βdx=31β[x]01β=31β
Myβ=β«β«xp(x,y)dA=β«x=01ββ«y=01βxydydx=β«x=01βx[2y2β]01βdydx=21ββ«x=01βxdx=21β[2x2β]01β=41β
And,
Mass,m is given by,
m=β«β«p(x,y)dA=β«x=01ββ«y=01βydydx=β«x=01β[2y2β]01βdydx=21ββ«x=01βdx=21β[x]01β=21β
Let (x,y) be the center of mass.
y=mMyββ=21β41ββ=21βx=mMxββ=21β31ββ=32β
Thus, center mass is (32β,21β).
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