Q. 40 - 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 $18\mathrm{\pi }$

Step 1. Given Information

The given figure is

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

Step 2. Finding Volume

$$\begin{gathered} V=\mathrm{\pi }{\int }_0^3\left[6-{\left(2+y\right)}^2-1^2\right]dy \\ =\mathrm{\pi }{\int }_0^3\left[\left(6-(2+\mathrm{y}\right){)}^2-1\right]\mathrm{dy} \\ =\mathrm{\pi }{\int }_0^3\left[15+{\mathrm{y}}^2-8\mathrm{y}\right]\mathrm{dy} \\ =\mathrm{\pi }{\left[15\mathrm{y}+\frac{{\mathrm{y}}^3}{3}-4{\mathrm{y}}^2\right]}_0^3 \\ =\mathrm{\pi }\left(45+9-36\right)=18\mathrm{\pi } \end{gathered}$$