Question

Use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ r=2 \theta, \quad 0 \leq \theta \leq \frac{\pi}{2} $$

Short answer

The length of the curve is approximately the result from the integration in step 3, rounded to two decimal places.

Step-by-step solution

Graphing the Curve

Input the polar equation \(r = 2\theta\) into the graphing utility. Make sure to set the range of \(\theta\) from 0 to \(\frac{\pi}{2}\). This will produce the graph of the curve over the specified interval.

Understanding the length of a polar curve

The length of a polar curve is given by the formula \[L= \int_a^b \sqrt{r^2 + (dr/d\theta)^2} d\theta\]. In our case, \(r=2\theta\), hence, we first need to calculate \(dr/d\theta\) which gives us 2.

Performing the Integration

Input the integral \(L= \int_0^{pi/2} \sqrt{(2\theta)^2 + 2^2} d\theta\) into the graphing utility’s numerical integration function. Round the answer to two decimal figures.