`y=x^2, y=x^3` 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

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,

`M_x=rhoint_a^b 1/2([f(x)]^2-[g(x)]^2)dx`


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



Now we are given `y=x^2,y=x^3`

Refer the attached image. Plot in red color is of `y=x^2` and blue color is of `y=x^3`

Curves intersect at (1,1)

Now let's evaluate the area of the region,






Now let's evaluate the moments about the x- and y-axes,

`M_x=rhoint_0^1 1/2[(x^2)^2-(x^3)^2]dx`














Plug in the values of `M_y` and `A` ,







The coordinates of the center of mass are `(3/5,12/35)`


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