Q. 56 - Calculus

Question

Consider the region between the graph of f(x) = 1 − cos x and the x-axis on [0,π]. For each line of rotation given in Exercises 55–58, write down definite integrals that represent the volume of the resulting solid and then use a calculator or computer to approximate the integrals.

Step-by-Step Solution

Answer

The volume of the resulting solid is V=43.753.

Step 1. Given

The graph of f(x) = 1 − cos x and the x-axis on [0,π].  

Step 2. Calculation

To determine the volume of solid of revolution, rotated around vertical line, express the curve as inverse function.

$$\begin{gathered} f\left(x\right)=1-\cos x \\ y=1-\cos x \\ \cos x =1-y \\ x={\cos }^{-1}(1-y) \\ p\left(y\right)={\cos }^{-1}(1-y) \end{gathered}$$

Use the definition of function to determine the y-interval...

For the x-interval of [0,7], the corresponding interval of y-variable will be [0,2]

For the washer, the external radius of each washer is and internal radius of each washer is given as p(y)

The volume of a washer is given by the integral

$V=\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}\left({\left(\mathrm{R}\left(\mathrm{y}\right)\right)}^2\right)-{\left(\mathrm{r}\left(\mathrm{y}\right)\right)}^2)\mathrm{dy}$

Use this definition and the values determined above to determine the volume of solid of

revolution.

$$\begin{gathered} V=\mathrm{\pi }{\int }_0^2\left({\left(\mathrm{\pi }\right)}^2-{\left(\mathrm{p}\left(\mathrm{y}\right)\right)}^2\right)\mathrm{dy} \\ \mathrm{V}=\mathrm{\pi }{\int }_0^2\left({\left(\mathrm{\pi }\right)}^2-{\left({\cos }^{-1}(1-\mathrm{y})\right)}^2\right)\mathrm{dy} \\ \mathrm{V}=\mathrm{\pi }{\int }_0^2\left({\left(\mathrm{\pi }\right)}^2-({\left({\cos }^{-1}(1-\mathrm{y})\right)}^2\right)\mathrm{dy} \\ \mathrm{V}=\mathrm{\pi }(4+{\mathrm{\pi }}^2) \\ =43.573 \end{gathered}$$