Answer in Calculus for naji #236118

A triangular lamina in the xy -plane such that its vertices are (0,0), (0,1) and (1,0). Suppose that the density function of the lamina is defined by p(x,y)=30xy gram per cubic centimetre. What is the total mass of the lamina and the center of gravity. The moment about the y-axis of the lamina.the moment about the x axis of the lamina

Let “A=(0,1), B=(1, 0), O=(0,0).”

Line OA: “x=0, 0leq yleq1”

LineAB: “y=-x+1, 0leq xleq1”

Line OB: “y=0, 0leq xleq 1”

Given “rho(x,y)=30xy”

Find the mass of the lamina

“m=iint_Drho(x,y)dA=displaystyleint_{0}^1displaystyleint_{0}^{1-x}30xydydx=”“=30displaystyleint_{0}^1xbig[{y^2over 2}big]begin{matrix}n 1-x \n 0 nend{matrix}dx=15displaystyleint_{0}^1(x-2x^2+x^3 )dx=”“=15big[{x^2 over 2}-{2x^3 over 3}+{x^4 over 4}big]begin{matrix}n 1 \n 0nend{matrix}={5 over 4} (units of mass)”

Mass of the lamina is “dfrac{5}{4}” units of mass.

The moment of the lamina about the x-axis

“M_x=iint_Dyrho(x,y)dA=30displaystyleint_{0}^1displaystyleint_{0}^{1-x}xy^2dydx=”

“=30displaystyleint_{0}^1xbig[{y^3over 3}big]begin{matrix}n 1-x \n 0 nend{matrix}dx”

“=10displaystyleint_{0}^1(x-3x^2+3x^3-x^4 )dx”

“=10big[{x^2 over 2}-{3x^3 over 3}+{3x^4 over 4}-{x^5 over 5}big]begin{matrix}n 1 \n 0nend{matrix}”

“=5-10+dfrac{15}{2}-2={1 over2}”

The moment of the lamina about the y-axis

“M_y=iint_Dxrho(x,y)dA=displaystyleint_{0}^1displaystyleint_{0}^{1-x}30x^2ydydx”

“=30displaystyleint_{0}^1x^2big[{y^2over 2}big]begin{matrix}n 1-x \n 0 nend{matrix}dx”

“=15displaystyleint_{0}^1(x^2-2x^3+x^4 )dx”

“=15big[{x^3 over 3}-{2x^4 over 4}+{x^5 over 5}big]begin{matrix}n 1 \n 0nend{matrix}”

“=5-dfrac{15}{2}+3=dfrac{1}{2}”

“M_x=dfrac{1}{2}, M_y=dfrac{1}{2}”

Find the coordinates of the center of mass

“bar{x}={M_yover m}=dfrac{dfrac{1}{2}}{dfrac{5}{4}}=dfrac{2}{5}”

“bar{y}={M_xover m}=dfrac{dfrac{1}{2}}{dfrac{5}{4}}=dfrac{2}{5}”

Center of gravity is“bigg(dfrac{2}{5},dfrac{2}{5}bigg).”