Q. 57 - 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=18.440.

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.] the corresponding interval of y-variable will be [0.2]

For the disk the radius is $\mathrm{\pi }-\mathrm{p}\left(\mathrm{y}\right)$

The volume of a disk is given by the integral

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

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

$$\begin{gathered} V=\mathrm{\pi }({\mathrm{\pi }}^2-4) \\ \mathrm{V}=18.440 \end{gathered}$$