Question

A thin uniform sheet of metal of uniform thickness is cut into the shape bounded by the line $x=a$ and $y=\pm k{x}^2$, as shown. Find the coordinates of the centre of mass.

Answer 1

According to the given question

$=\frac{1}{a}{\int }_0^{a}x(y_1-y_2)dx=\frac{1}{a}{\int }_0^{a}x(k{x}^2-(-k{x}^2\left)\right)dx=\frac{2k}{a}{\int }_0^a{x}^{3}dx=\frac{2k}{a}\times \frac{a^4}{4}=\frac{{ka}^3}{2}$