Question

In Problems 17 to 30, for the curve $\mathit{y}\mathbf{=}\sqrt{\mathbf{x}}$, between$\mathbf{x}\mathbf{=}\mathbf{0}$and$\mathit{x}\mathbf{=}\mathbf{2}$,

find:

The centroid of the volume.

Short answer

The centroid of the volume is$\left(\overline{x},\overline{y},\overline{z}\right)=\left(\frac{4}{3},0,0\right)$ .

Step-by-step solution

Definition of Centroid of volume

The geometric center of a particular body having a specific shape is known as the centroid of volume. In the case of uniform density, the center of mass lies on the centroid of volume, and also the center of gravity lies on it.

Representation of the area bounded by the curve

Draw the bounded region for the curve $y=\sqrt{x}$, between $x=0$and $x=2$.

 Step 3: Calculation of the value of x

If the surface be rotated around the x- axis, then only $\overline{x}$remains, and $\overline{y}=\overline{z}=0$.

$$\begin{gathered} \overline{x}=\frac{1}{V}{\int }_0^2\mathrm{\pi }{\left(\mathrm{f}\left(\mathrm{x}\right)\right)}^2\mathrm{xdx} \\ =\frac{1}{2\mathrm{\pi }}{\int }_0^2{\mathrm{\pi x}}^2\mathrm{xdx} \\ {\overline{)=\frac{1}{6}{\mathrm{x}}^3}}_0^2 \\ =\frac{4}{3} \end{gathered}$$

Thus, the centroid of the volume $\left(\overline{x},\overline{y},\overline{z}\right)=\left(\frac{4}{3},0,0\right)$.