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. $$ r=\theta, \quad 0 \leq \theta \leq \pi $$

Short answer

The approximate surface area of the revolution about the polar axis is \(31.00\) to two decimal places.

Step-by-step solution

Understanding the Polar Function

The given function is in polar coordinates where the radius is increasing linearly with \(\theta\) from 0 to \(\pi\). When \(\theta = 0, r = 0\) and when \(\theta = \pi, r = \pi\). In polar coordinates, the polar axis is the positive x-axis, and the angle \(\theta\) is measured counter-clockwise from the polar axis. This is the curve we will rotate around the polar axis.

Visualizing the Revolution

Use a graphing utility to graph the function and visually comprehend the revolution about the polar axis. You'll see that the revolution results in a cone with a base radius and height of \(\pi\) in the polar plane.

Calculating the Surface Area

The formula for the lateral surface area \(A\) of a cone with base radius \(r\) and slant height \(l\) is \(A = \pi rl\). Here, the radius \(r\) and the slant height \(l\) both equal \(\pi\). So, substitute \(r = \pi\) and \(l = \pi\), the expression becomes \(A = \pi.(\pi).\(\pi\) = \(\pi^3\).

Approximating to Two Decimal Places

As the last step, you are asked to approximate the result to two decimal places. The approximated value of surface area \(A\approx 31.00\) to two decimal places.