`y = 1/2x , y = 0 , x =2` Find the x and y moments of inertia and center of mass for the laminas of uniform density `p` bounded by the graphs of the equations.
For an irregularly shaped planar lamina of uniform density `(rho)` , bounded by graphs `y=f(x),y=g(x)` and `a<=x<=b` ,the mass (m) of this region is given by,
`m=rhoA` ,where A is the area of the region.
The moments about the x and y-axes are given by the formula,
The coordinates of the center of mass `(barx,bary)` are given by,
We are given `y=1/2x,y=0,x=2`
The attached image shows the region bounded by the functions and the limits of integration,
Let's evaluate the area of the region,
Evaluate the integral by applying power rule,
Now let's evaluate the moments about the x and y-axes,
Apply the power rule,
Apply power rule,
Now let's find the coordinates of the center of mass,
The coordinates of the center of mass are `(4/3,1/3)`