Q. 55 - 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=14.804.

Step 1. Given

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

Step 2. Calculation

The given region bounded by f(x)=1-cos x and x-axis, between the interval [0,]is rotated around the x-axis, y=0. The objective is to determine the volume of solid of revolution using definite integrals.

The radius of each disk is given as f(x).

$V=\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}(\mathrm{f}{\left(\mathrm{x}\right))}^2\mathrm{dx}$

The total volume of solid is determined by

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

revolution.

$V=\mathrm{\pi }{\int }_{\mathrm{a}}^{\mathrm{b}}{(1-\cos x)}^2\mathrm{dx}$