Q. 38 - 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 is 393.64 cubic unit.

Step 1. Given Information

The given figure is 

Step 2. Finding Volume

The external radius of each washer is 3 and internal radius is $-3-f\left(x\right)$.

The volume of solid of revolution is 

$$\begin{gathered} V=\mathrm{\pi }{\int }_2^5\left(3^2-(-3-\mathrm{f}{\left(\mathrm{x}\right))}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(3^2-(-3-{(\mathrm{x}-2))}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(9+{(3+(\mathrm{x}+2)}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(9+x^2+4x+4\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(13+x^2+4x\right)\mathrm{dx} \\ =\mathrm{\pi }{\left[13x+\frac{x^3}{3}+2x^2\right]}_2^5 \\ =\mathrm{\pi }\left(\left(65+\frac{125}{3}+50\right)-\left(26+\frac{8}{3}+8\right)\right) \\ =\mathrm{\pi }\left(115-34+\frac{125}{3}+\frac{8}{3}\right)=\mathrm{\pi }\left(81+44.3\right) \\ =125.3\mathrm{\pi } \\ =393.64 \end{gathered}$$