Question

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

The mass of the solid of revolution if the density (mass per unit volume) is $\left|\mathrm{xyz}\right|$.

Short answer

The mass of the solid of revolution if the density is2.

Step-by-step solution

Definition of solid of revolution

The solid that is obtained on rotating a figure which is plane in nature around a particular axis of the straight line is known as solid of revolution.

Representation of the area bounded by the curve

Draw the bounded region for the curve.

Determination of the mass

The mass contained in each of the$x\ge 0$octants is equal. So,it can be integrated over the$y\ge 0$,$z\ge 0$octant and just multiply the result by 4.

Determine the mass of the volume by Integrating over z .

$$\begin{gathered} M=4{\int }_0^{2}xdx\left[{\int }_0^{f\left(x\right)}ydy{\int }_0^{\sqrt{f^2\left(x\right)-y^2}}zdz\right] \\ =4{\int }_0^{2}xdx\left[{\int }_0^{\sqrt{x}}ydy{\int }_0^{\sqrt{x-y^2}}zdz\right] \\ =4{\int }_0^{2}xdx\left[{\int }_0^{\sqrt{x}}ydy{\overline{)\frac{1}{2}{z}^2}}_0^{\sqrt{x-y^2}}\right] \\ =2{\int }_0^{2}xdx\left[{\int }_0^{\sqrt{x}}ydy\left(x-y^2\right)\right] \end{gathered}$$

Determination of the mass of the solid of revolution

Integrate over x and y .

$$\begin{gathered} M=2{\int }_0^{2}xdx{\overline{)\left[\frac{1}{2}x{y}^2-\frac{1}{4}{y}^4\right]}}_0^{\sqrt{x}} \\ =2{\int }_0^{2}xdx\frac{x^2}{4} \\ =2{\int }_0^2\frac{x^3}{2}dx \\ ={\overline{)\frac{1}{8}{x}^4}}_0^2 \\ =2 \end{gathered}$$

Thus, mass of the solid of revolution if the density is 2.