Q. 31 - Calculus

Question

Consider the region between the graph of $f\left(x\right)=\frac{3}{x}$ and the x-axis on [1,3]. For each line of rotation given in Exercises 31– 34, use definite integrals to find the volume of the resulting solid.

Step-by-Step Solution

Answer

The required volume is $6\mathrm{\pi }$

Step 1. Given Information

The given figure is     

Step 2. Calculation

The radius of each disk is given as $f\left(x\right)$.

The total volume of solid is determined by,

$$\begin{gathered} V=\mathrm{\pi }{\int }_1^3{\left(\frac{3}{\mathrm{x}}\right)}^2\mathrm{dx} \\ =\mathrm{\pi }{\int }_1^3\frac{9}{{\mathrm{x}}^2}\mathrm{dx} \\ =9\mathrm{\pi }{\int }_1^3{\mathrm{x}}^{-2}\mathrm{dx} \\ =9\mathrm{\pi }{\left[\frac{{\mathrm{x}}^{-1}}{-1}\right]}_1^3 \\ =-9\mathrm{\pi }{\left[\frac{1}{\mathrm{x}}\right]}_1^3 \\ =-9\mathrm{\pi }\left(\frac{1}{3}-1\right) \\ =6\mathrm{\pi } \end{gathered}$$