Lab 5: Arclength

Student Resources

Lab 5: Arclength - part 2

Continue your work from Lab 4 to approximate the arclength of the graph of \(f(x) =\sin x\) between \(x = 0\) and \(x =\frac{\pi}{2}\).
1. Use summation notation to express a general approximation for n subintervals of size \(\Delta x\).
2. Explain why this is not a Riemann sum.
3. Recall that the Mean Value Theorem (MVT) says if a function is differentiable on an interval \([a,b]\) then there is a point c in this interval such that \[ f'(c)(b-a) = f(b) - f(a)\] Use the MVT to convert your answer in Part a to a Riemann sum.
Now that you have your approximation expressed as a Riemann sum, you can take a limit as \(n\to \infty\).
1. Express the limit of your answer to Question 1 as a definite integral.
2. Is this an integral you can compute using the Fundamental Theorem of Calculus? Why?
3. Use Simpson's Rule with \(n=2\) to approximate this integral.
4. The \(4^\text{th}\) derivative of \(\sqrt{1 + \cos^2 x}\) varies between \(-7\) and 2.04 on the interval \(\left[0,\frac{\pi}{2}\right]\). Use this to find an error bound for your approximation in Part c.
Write an integral to express the arclength of the graph for each of the following functions on the given interval. Use a midpoint rule with \(n=10\) to approximate the integral.
1. \(g(x) = \sin^2 x\) from \(x = 0\) to \(x = \pi/2\).
2. \(y(x) = \ln(2x)\) from \(x = 1/2\) to \(x = e/2\).
3. \(r(x) = \tan x\) from \(x = -\pi/4\) to \(x = \pi/4\).
4. \(p(x) = \arcsin x\) from \(x = -1/2\) to \(x = 1/2\).
5. \(h(x) = e^{x/2}\) from \(x = -1\) to \(x = 1\).
6. \(d(x) = \cos x\) from \(x = -\pi\) to \(x = \pi\).