find the centre of mass of a thin plate of constant density covering the region bounded by the parabola y=25-x^2 and the x axis.

A thin strip (thickness dx) of the plate, along the vertical line with abscissa x, has mass
y dx = delta*(25 - x^2) dx
(The symbol I've seen used for this most often is rho, which looks like a curly slanted p)
Hence the mass of the plate is
Int[-5 to 5] (25-x^2) dx * delta
= [25x - (x^3)/3] [-5 to 5] * delta
= 500delta/3
The x coordinate of the centre of mass is
Int[-5 to 5] x(25-x^2)dx * delta divided by (500delta/3)
= [(25x^2)/2 - (x^4)/4] [-5 to 5] *3/500
= 0 as we would expect.
The y coordinate of the centre of mass of each of these strips is, of course, (25-x^2)/2, and so the y coordinate of the centre of mass of the plate is
Int[-5 to 5] (((25 - x^2)^2)/2) dx *3/500
= [625x/2 - (25x^3)/3 + (x^5)/10] [-5 to 5] *3/500
= 2*(625*5/2-25*(5^3)/3 + (5^5)/10)*3/500
= 10
Hence the centre of mass is (0, 10)
Thanks & Regards,
Nirmal Singh
Askiitians Faculty