Lab 4: Arclength

Lab 4: Arclength - part 1

Draw a full page graph showing:
1. The graph of \(f(x) = \sin x\) between \(x = 0\) and \(x = \frac{\pi}{2}\).
2. Points on the graph for \(x = 0\), \(x = \frac{\pi}{6}\), \(x = \frac{\pi}{3}\), and \(x = \frac{\pi}{2}\) labeled with the exact values of their \((x,y)\)-coordinates. Note that this partitions the domain into 3 equal subintervals.
3. Three secant lines connecting the points in Part b labeled with their lengths.
The sum of the lengths of the three secant lines from Question 1 can be used to approximate the arclength of the graph of \(f(x) =\sin x\) between \(x = 0\) and \(x = \frac{\pi}{2}\).
1. What is this approximation?
2. Is it an underestimate or overestimate? Explain how you know.
Find a way to compute the approximation in Question 2 using a sum(seq(... command.

You will then generalize this command to other partitions of the interval \(\frac{\pi}{2}\).
1. Write down your sum(seq(... command for \(n =3\), \(\Delta x = \frac{\pi}{6}\) and verify that it works by making sure it gives the same answer you got in Question 2.
2. Modify and evaluate your sum(seq(...command for \(n =4\), \(\Delta x = \frac{\pi}{8}\).
3. Modify and evaluate your sum(seq(... command for \(n =50\), \(\Delta x = \frac{\pi}{100}\) .
4. Modify and evaluate your sum(seq(... command for \(n =500\), \(\Delta x = \frac{\pi}{1000}\).
Use summation notation to express a general approximation for \(n\) subintervals of size \(\Delta x\). Explain why your sum is not a Riemann sum.