Lab 5: Arclength - part 2


Instructions: Work with your group on all questions in this lab. We encourage you to collaborate both in and out of class, but you must write up your responses individually. Your work must be neat and include sufficient exposition to make the solution clear to another student who has not seen the assignment (for example, a sequence of equations without explanation will most likely receive zero credit). Pay particular attention to places where explanations using multiple representations are requested, and explicitly discuss the connections between your explanations using different representations. Draw a picture of a torus in the lower right corner of your first page for five extra credit points. Type or write all of your work legibly on 8\(\frac{1}{2}\)''×11'' paper with no spiral fringe, at least one-inch margins on all sides free of writing except your name, date, and assignment number, and staple all pages together. 

 

  1. 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.

  2. 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.

  3. 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 \(x = \pi\).