To determine the center of gravity (centroid) of the paraboloid surface: ### Assumptions: - The paraboloid is symmetric about the \( z \)-axis. - The paraboloid is generated by rotating a parabola about the \( z \)-axis. - The paraboloid has a height \( h \) and extends over a base with side length \( 2a \). --- ### Coordinates: - The paraboloid is centered at the origin. - The general equation of a paraboloid: \( z = \frac{\alpha}{a^2}(x^2 + y^2) \). --- ### Approach: - Since the paraboloid is symmetric about the \( z \)-axis, the centroid along the \( x \) and \( y \) axes will be at zero: \[ \bar{x} = , \quad \bar{y} = \] - The \( z \)-coordinate of the centroid can be found using the volume integral: \[ \bar{z} = \frac{1}{V} \iiint z \, dV \] where \( V \) is the volume of the paraboloid. --- ### Result: - For a paraboloid of revolution, the centroid along \( z \) is located at: \[ \bar{z} = \frac{h}{2} \] - Because of the symmetry: \[ \boxed{ \begin{aligned} \text{Center of gravity } (\bar{x}, \bar{y}, \bar{z}) &= (, , \frac{h}{2}) \end{aligned} } \] --- ### Final answers: - \( \text{Location of } \left(\frac{\bar{x}}{a}\right) \) = **** - \( \text{Location of } \left(\frac{\bar{y}}{a}\right) \) = **** - \( \text{Location of } \left(\frac{\bar{z}}{h}\right) \) = **\(\frac{1}{2}\)** --- **Note:** The values are normalized by \( a \) and \( h \), respectively.