Q. 50 - Calculus

Question

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{region} \mathrm{between} \mathrm{the} \mathrm{graphs} \mathrm{of} \mathrm{f} \left(\mathrm{x}\right) = {\mathrm{x}}^2 \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = 2\mathrm{x} \mathrm{on} [0, 2]. \\ \mathrm{For} \mathrm{each} \mathrm{line} \mathrm{of} \mathrm{rotation} \mathrm{given} \mathrm{in} \mathrm{Exercises} 47–50, \mathrm{use} \mathrm{definite} \mathrm{integrals} \mathrm{to} \mathrm{find} \\ \mathrm{the} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{resulting} \mathrm{solid}. \end{gathered}$$

Step-by-Step Solution

Answer

$\mathrm{The} \mathrm{volume} \mathrm{of} \mathrm{solid} \mathrm{of} \mathrm{revolution}, \mathrm{V}=\frac{16}{3}\mathrm{\pi }$

Step 1. Given information is:

$$\begin{gathered} \mathrm{The} \mathrm{given} \mathrm{region} \mathrm{bounded} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)=2\mathrm{x} , \mathrm{between} \mathrm{the} \mathrm{interval} [0,2], \\ \mathrm{is} \mathrm{rotated} \mathrm{around} \mathrm{vertical} \mathrm{line}, \mathrm{x}=3. \end{gathered}$$

Step 2. Determining Inverse function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2 \\ \mathrm{y}={\mathrm{x}}^2 \\ \mathrm{x}=\sqrt{\mathrm{y}} \\ \mathrm{p}\left(\mathrm{y}\right)=\sqrt{\mathrm{y}} \\ \mathrm{Determining} \mathrm{the} \mathrm{inverse} \mathrm{of} \mathrm{other} \mathrm{function} \mathrm{also}, \\ \mathrm{g}\left(\mathrm{x}\right)=2\mathrm{x} \\ \mathrm{y}=2\mathrm{x} \\ \mathrm{x}=\frac{\mathrm{y}}{2} \\ \mathrm{q}\left(\mathrm{y}\right)=\frac{\mathrm{y}}{2} \end{gathered}$$

Step 3. Determing y-interval

$\mathrm{For} \mathrm{the} \mathrm{x}-\mathrm{interval} \mathrm{of} [0,2], \mathrm{the} \mathrm{corresponding} \mathrm{interval} \mathrm{of} \mathrm{y}-\mathrm{variable} \mathrm{will} \mathrm{be} [0,4]$

Step 4. Determining Volume of Solid of Revolution

$$\begin{gathered} \mathrm{For} \mathrm{the} \mathrm{washer}, \mathrm{the} \mathrm{external} \mathrm{radius} \mathrm{of} \mathrm{each} \mathrm{washer} \mathrm{is} 3-q\left(\mathrm{y}\right) \mathrm{and} \mathrm{internal} \mathrm{radius} \mathrm{of} \\ \mathrm{each} \mathrm{washer} \mathrm{is} \mathrm{given} \mathrm{as} 3-p\left(\mathrm{y}\right). \\ \mathrm{The} \mathrm{volume} \mathrm{of} \mathrm{washer} \mathrm{is} \mathrm{given} \mathrm{by}: \\ \mathrm{V}=\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}\left({\left(\mathrm{R}\left(\mathrm{y}\right)\right)}^2-{\left(\mathrm{r}\left(\mathrm{y}\right)\right)}^2\right)\mathrm{dy} \\ \mathrm{Using} \mathrm{this} \mathrm{definition} \mathrm{to} \mathrm{determine} \mathrm{the} \mathrm{volume} \mathrm{of} \mathrm{solid} \mathrm{of} \mathrm{revolution}, \\ \mathrm{V}=\mathrm{\pi }{\int }_0^4\left({\left(3-\mathrm{q}\left(\mathrm{y}\right)\right)}^2-{\left(3-\mathrm{p}\left(\mathrm{y}\right)\right)}^2\right)\mathrm{dy} \\ \mathrm{V}=\mathrm{\pi }{\int }_0^4\left({\left(3-\frac{\mathrm{y}}{2}\right)}^2-{\left(3-\sqrt{\mathrm{y}}\right)}^2\right)\mathrm{dy} \\ \mathrm{V}=\mathrm{\pi }{\int }_0^4\left(\left(9-3y+\frac{{\mathrm{y}}^2}{4}\right)-\left(9-6\sqrt{\mathrm{y}}+y\right)\right)\mathrm{dy} \\ \mathrm{V}=\mathrm{\pi }{\int }_0^4\left(-4y+\frac{{\mathrm{y}}^2}{4}+6\sqrt{\mathrm{y}}\right)\mathrm{dy} \\ \mathrm{V}=\mathrm{\pi }{\left[-\frac{4y^2}{2}+\frac{{\mathrm{y}}^3}{12}+6\frac{y^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right]}_0^4 \\ \mathrm{V}=\mathrm{\pi }\left[\left(-32+\frac{16}{3}+32\right)-\left(0\right)\right] \\ \mathrm{V}=\frac{16}{3}\mathrm{\pi } \end{gathered}$$