Q. 56 - Calculus

Question

Consider the region between the graph of $f\left(x\right)=1-\cos x$ 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 solid is

Step 1: Set up using appropriate method

For \(f(x) = 1 - \cos x\), the volume of revolution uses the disk method if rotating about the x-axis: \(V = \pi\int (1-\cos x)^2\,dx\).

Step 2: Expand and integrate

\((1-\cos x)^2 = 1 - 2\cos x + \cos^2 x = 1 - 2\cos x + \frac{1+\cos 2x}{2} = \frac{3}{2} - 2\cos x + \frac{\cos 2x}{2}\). Integrate term by term over the specified interval.