A paraboloid shaped solid object has equation ${\text{y\hspace{0.17em}=\hspace{0.17em}Kx}}^{\text{2}}$  about y-axis as shown in figure. If the height of the body is ‘h’ then the position of centre of mass from origin (Assume density to be uniform throughout) is

Given, parabolic equation

${\text{y\hspace{0.17em}=\hspace{0.17em}kx}}^{\text{2}}\Rightarrow {\text{\hspace{0.17em}\hspace{0.17em}x}}^{\text{2}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{y}}{\text{k}}$

Let $\text{ρ}$ be the volume density of the paraboloid.

Consider an infinitesimal disc of radius x, width dy, mass dm at a distance y from O.

$\Rightarrow \text{\hspace{0.17em}dm\hspace{0.17em}=\hspace{0.17em}ρ}\left({\text{πx}}^{\text{2}}\right)\text{\hspace{0.17em}dy\hspace{0.17em}=\hspace{0.17em}}\pi \rho \left(\frac{y}{k}\right)\text{\hspace{0.17em}dys}$

$\Rightarrow {\text{\hspace{0.17em}y}}_{\text{COM}}=\frac{{\displaystyle \int \text{ydm}}}{{\displaystyle \int \text{dm}}}$

$\text{=\hspace{0.17em}\hspace{0.17em}}\frac{{\displaystyle {\int }_{\text{o}}^{\text{h}}\text{πρ\hspace{0.17em}(}\frac{{\text{y}}^{\text{2}}}{\text{k}}\text{dy)}}}{{\displaystyle {\int }_{\text{o}}^{\text{h}}\text{πρ\hspace{0.17em}(}\frac{\text{y}}{\text{k}}\text{)\hspace{0.17em}dy}}}$$\text{=\hspace{0.17em}\hspace{0.17em}}\frac{{\text{[}\frac{{\text{y}}^{\text{3}}}{\text{3}}\text{]}}^{\text{h}}}{{\text{[}\frac{{\text{y}}^{\text{2}}}{\text{2}}\text{]}}_{\text{o}}^{\text{h}}}$

$\Rightarrow {\text{\hspace{0.17em}\hspace{0.17em}y}}_{\text{COM}}\text{\hspace{0.17em}=\hspace{0.17em}}\frac{\text{2h}}{\text{3}}$