Q. 32 - 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 $9.26\mathrm{\pi }$

Step 1. Given Information

The given figure is     

Step 2. Calculation

The external radius of each washer is $1+f\left(x\right)$ and internal radius is 1.

The volume of a washer is determined by,

$$\begin{gathered} V=\mathrm{\pi }{\int }_1^3\left({\left(1+\mathrm{f}\left(\mathrm{x}\right)\right)}^2-1^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_1^3\left({\left(1+\frac{3}{\mathrm{x}}\right)}^2-1^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_1^3\left(1+\frac{9}{{\mathrm{x}}^2}+\frac{6}{\mathrm{x}}-1\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_1^3\left(-9{\mathrm{x}}^{-2}+\frac{6}{\mathrm{x}}\right)\mathrm{dx} \\ =\mathrm{\pi }{\left[\frac{9}{\mathrm{x}}+6\ln \left(\mathrm{x}\right)\right]}_1^3 \\ =\mathrm{\pi }\left(\left(\frac{9}{3}+6\ln 3\right)-\left(\frac{1}{3}+6\ln 1\right)\right) \\ =\mathrm{\pi }\left(3+6\ln 3-\frac{1}{3}-0\right) \\ =9.26\mathrm{\pi } \end{gathered}$$