Q. 39 - 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 volume of the solid is $9\mathrm{\pi }$

Step 1. Given Information

The given figure is

Here $$\begin{gathered} f\left(x\right)=x-2=y \\ x=y+2 \end{gathered}$$

Step 2. Finding Volume

$$\begin{gathered} V=\mathrm{\pi }{\int }_0^3{\left(-y\right)}^{2}dy \\ V=\mathrm{\pi }{\int }_0^3{y}^{2}dy \\ V=\frac{y^3}{3} \\ V=\frac{3^3}{3}\mathrm{\pi }=9\mathrm{\pi } \end{gathered}$$