Question

Use the integration capabilities of a graphing utility to approximate to two decimal places the area of the surface formed by revolving the curve about the polar axis. $$ r=4 \cos 2 \theta, \quad 0 \leq \theta \leq \frac{\pi}{4} $$

Short answer

The exact final result depends on the result from the integration. After integrating, one should ensure to round the surface area to two decimal places.

Step-by-step solution

Conversion of the polar equation and setting up the integral

First, plug the given equation \(r = 4\cos 2\theta\) and the range of \(\theta\) into the surface area formula. Also, because the curve is revolved around the polar axis, \(\sin \theta = 1\). So the equation to calculate the surface area becomes \(A = 2\pi \int_{0}^{\pi/4} 4\cos(2\theta) \, d\theta\).

Performing the integration

Now that we have set up the formula, the next step is to perform the actual integration. This requires knowledge of integration methods and is facilitated by using a graphing utility.

Getting the numerical result

After evaluating the integral either manually or with a graphing utility, round the final result to two decimal places.