Answer to Question #193303 in Calculus for Sourav

Question

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the 

curves

y = 4x² and x = 4

Answer 1

$\text{Given, density} \delta(x,y)=y\text{ and bounded by }y=4x^2and x=4.\newline \text{mass},m=\int\int\delta dA=\int\int y dxdy=\int_{x=0}^4\int_{y=0}^{x^2}y dxdy=\frac{1}{2}\int_{x=0}^4x^4dxdy=\frac{1024}{10}=102.4\newline \text{Now, calculate moments}\newline M_x=\int\int y\delta dA=\int\int y^2dxdy=\int_{x=0}^4\int_{y=0}^{x^2}y^2dxdy=\frac{1}{3}\int_{x=0}^4x^6dxdy=\frac{16384}{21}=780.19\newline M_y=\int\int x\delta dA=\int\int xydxdy=\int_{x=0}^4\int_{y=0}^{x^2}xydxdy=\frac{1}{2}\int_{x=0}^4 x^5dxdy=\frac{4096}{12}=341.33\newline \text{Let (X, Y) be the center of mass.}\newline X=\frac{M_y}{m}=\frac{341.33}{102.4}=3.33\newline Y=\frac{M_x}{m}=\frac{780.19}{102.4}=7.62\newline \text{Center of mass is} (3.33, 7.62).\newline \text{Therefore, center of gravity is }(3.33, 7.62).$