Question

Repeat Problem 23a for the paraboloid$\frac{\mathbf{z}}{\mathbf{c}}\mathbf{=}\frac{{\mathbf{x}}^{\mathbf{2}}}{{\mathbf{a}}^{\mathbf{2}}}\mathbf{+}\frac{{\mathbf{y}}^{\mathbf{2}}}{{\mathbf{b}}^{\mathbf{2}}}\mathbf{}\mathbf{}\mathbf{0}\mathbf{<}\mathbf{z}\mathbf{<}\mathbf{c}\mathbf{.}$

Short answer

The centroid of the solid paraboloid is $\left(\overline{x},\overline{y},\overline{z}\right)=\left(0,0,\frac{2}{3}c\right)$

Step-by-step solution

 Step 1: Given Condition

The paraboloid is $\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^{2.}}0<z<c$

Concept of centroid

The centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure.

 Step 3: Calculate the volume

The volume is given by the following formula,

$$\begin{gathered} V={\int }_x{\int }_y{\int }_{z}dxdydz \\ ={\int }_x{\int }_y{\int }_{z}dx\text{'}dy\text{'}dz\text{'}abc \\ =abc{\int }_{v}dV \\ =2\mathrm{\pi abc}{\int }_0^1\mathrm{dz}\text{'}\frac{1}{2}\mathrm{z}\text{'} \\ =\frac{\mathrm{\pi }}{2}\mathrm{abc} \end{gathered}$$

Thus,

$\overline{x}=\overline{y}=0$

Calculate the  z coordinate

Now, to calculate the value of $\overline{z}$ ,

$$\begin{gathered} V={\int }_x{\int }_y{\int }_{z}zdxdydz \\ =ab{c}^2{\int }_x{\int }_y{\int }_{z}dx\text{'}dy\text{'}dz\text{'}z\text{'} \\ =ab{c}^2{\int }_0^{2\mathrm{\pi }}d\theta {\int }_0^{1}dz\text{'}z{\int }_0^{\sqrt{z}}rdr \\ =2\mathrm{\pi abc}{\int }_0^{1}z\text{'}\mathrm{dz}\text{'}\frac{1}{2}\mathrm{z}\text{'} \\ ={\mathrm{\pi abc}}^2{\int }_0^{1}z{\text{'}}^{2}dz\text{'} \\ =\frac{\mathrm{\pi }}{3}c \\ \overline{z}=\frac{2}{3}c \end{gathered}$$

Hence, the centroid is .$\left(\overline{x},\overline{y},\overline{z}\right)=\left(0,0,\frac{2}{3}c\right)$.