Answer to Question #140559 in Calculus for Besmallah Yousefi

Question #140559

Find the mass and the center of mass of a triangular lamina with vertices (0.0), (1,0) and (0,2) if the density function is p=1+3x+y


1
Expert's answer
2020-10-27T19:17:45-0400

Mass,

M=0102(1+3x+y)dydx=01[y+3xy+y22]02dx=01(6x+4)dx=[3x2+4x]01=7M=\int_0^1\int_0^2(1+3x+y)\,dy\,dx\\ =\int_0^1[y+3xy+\frac{y^2}{2}]_0^2dx\\ =\int_0^1(6x+4)dx=[3x^2+4x]_0^1=7

Thus, mass is 7 unit.

Now,


Xcom=1M0102x(1+3x+y)dydx=1M01(6x2+4x)dx=4/7X_{com}=\frac{1}{M}\int_0^1\int_0^2x(1+3x+y)dydx\\ =\frac{1}{M}\int_0^1(6x^2+4x)dx\\ =4/7

Ycom=1M0102y(1+3x+y)dydx=1M01(6x+143)dx=23/21Y_{com}=\frac{1}{M}\int_0^1\int_0^2y(1+3x+y)dydx\\ =\frac{1}{M}\int_0^1(6x+\frac{14}{3})dx\\ =23/21


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