`x=4-y^2 , x=0` 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.

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gsarora17 eNotes educator| Certified Educator

Consider an irregularly shaped planar lamina of uniform density `rho` , bounded by graphs `x=f(y)` , `x=g(y)` and `c<=y<=d` . 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:

`M_x=rhoint_c^d y(f(y)-g(y))dy`

`M_y=rhoint_c^d 1/2([f(y)]^2-[g(y)]^2)dy`

The center of mass `(barx,bary)` is given by:



We are given `x=4-y^2` ,`x=0`  

Refer to the attached image. The plot of `x=4-y^2` is red in color.

Let's evaluate the area of the region,







`M_y=2rhoint_0^2 1/2(4-y^2)^2dy`








By symmetry, `M_x=0,bary=0`





The coordinates of the center of mass are `(barx,bary)` are `(8/5,0)`  

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