Question

Consider the functions

Answer 1

Assuming the region is of uniform density, compute the mass of the region and its moments.

If n > m, then in the interval [0, 1] we have f(x) ≤ g(x). This means the mass is

$M=\displaystyle\int_0^1(g(x)-f(x))\,\mathrm dx=\left(\frac{x^{m+1}}{m+1}-\frac{x^{n+1}}{n+1}\right)\bigg|_0^1=\frac1{m+1}-\frac1{n+1}$

The moments are

$M_x=\displaystyle\int_0^1\frac{g(x)^2-f(x)^2}2\,\mathrm dx=\frac12\left(\frac{x^{2m+1}}{2m+1}-\frac{x^{2n+1}}{2n+1}\right)\bigg|_0^1=\frac12\left(\frac1{2m+1}-\frac1{2n+1}\right)$

$M_y=\displaystyle\int_0^1x(g(x)-f(x))\,\mathrm dx=\int_0^1(x^{m+1}-x^{n+1})\,\mathrm dx=\left(\frac{x^{m+2}}{m+2}-\frac{x^{n+2}}{n+2}\right)\bigg|_0^1=\frac1{m+2}-\frac1{n+2}$

Then the centroid is

$(\overline x,\overline y)=\left(\dfrac{M_y}M,\dfrac{M_x}M\right)=\boxed{\left(\dfrac{(m+1)(n+1)}{(m+2)(n+2)},\dfrac{(m+1)(n+1)}{(2m+1)(2n+1)}\right)}$