Answer to Question #140559 in Calculus for Besmallah Yousefi

Question

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

Answer 1

Mass,

$$M=\int_0^1\int_0^2(1+3x+y)\,dy\,dx\\ =\int_0^1[y+3xy+\frac{y^2}{2}]_0^2dx\\ =\int_0^1(6x+4)dx=[3x^2+4x]_0^1=7$$

Thus, mass is 7 unit.

Now,

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

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