Q. 36 - Calculus

Question

Consider the region between the graph of $f\left(x\right)=x-2$ and the x-axis on [2,5]. For each line of rotation given in Exercises 35– 40, use definite integrals to find the volume of the resulting solid. 

Step-by-Step Solution

Answer

The required volume is  $9\mathrm{\pi }$

Step 1. Given Information

The given figure is     

Step 2. Calculation

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

Determine the volume of solid of revolution.

$$\begin{gathered} V=\mathrm{\pi }{\int }_2^5{\left(\mathrm{x}-2\right)}^2\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left({\mathrm{x}}^2-4\mathrm{x}+4\right)\mathrm{dx} \\ =\mathrm{\pi }{\left[\frac{{\mathrm{x}}^3}{3}-\frac{4{\mathrm{x}}^2}{2}+4\mathrm{x}\right]}_2^5 \\ =\mathrm{\pi }\left[\left(\frac{5^3}{3}-2{\left(5\right)}^2+4\left(5\right)\right)-\left(\frac{2^3}{3}-2{\left(2\right)}^2+4\left(2\right)\right)\right] \\ =\mathrm{\pi }\left[\left(\frac{125}{3}-50+20\right)-\left(\frac{8}{3}-8+8\right)\right] \\ =\mathrm{\pi }\left[\frac{125}{3}-30-\frac{8}{3}\right] \\ =9\mathrm{\pi } \end{gathered}$$