Find the center of mass of the parabolic lamina bounded by the curves y² = 2px and x = a, figure.

Obviously the center of mass lies on the x-axis. Therefore we need to determine x_cm only. Taking a strip of width dx for the element of mass we have

dm = σ 2ydx = 2σ √2pxdx

where σ is the mass per unit area. Therefore, substituting the expression of dm and changing the limits of integration we obtain

x_cm = 2σ∫a 0x √2pxdx/2σ∫a 0√2pxdx

⇒ x_cm = ∫a 0x^3/2 dx/∫a 0x^1/2 dx

⇒ x_cm = 3a/5