Lab 2: Integral approximations - part 1


Lab Preparation: Answer the following questions individually and bring your write-up to class.

  1. Under what conditions is the left-hand sum for a definite integral an underestimate? An overestimate? Why?:

  2. Under what conditions is the right-hand sum for a definite integral an underestimate? An overestimate? Why?

  3. The average of the left-hand sum and the right-hand sum for a definite integral represents the sum of the areas of trapezoids on each subinterval. Under what conditions do these trapezoids produce an underestimate? An overestimate? Why?

 

Lab Questions

 

Instructions: 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. 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.

 

 

In the first section of this lab, you will work with the function \(f(x) = 2^{-x}\). You will be able to compute the integral exactly using the fundamental theorem of calculus, which will allow you to see the sizes of the errors.

 

  1. Compute \(\int_{-1}^2 f(x) dx\).

  2. Find the approximations for \(\int_{-1}^2 f(x) dx\) with four subintervals using

    1. A left-hand sum, LEFT(4)

    2. A right-hand sum, RIGHT(4)

    3. The midpoint method, MID(4)

    4. The trapezoid method, TRAP(4) = \(\frac{1}{2}\)(LEFT(4) + RIGHT(4).

    5. Simpson's rule, SIMP(4)* = \(\frac{2}{3}\) MID(4) + \(\frac{1}{3}\) TRAP(4).

  3. Find the error for each of your approximations. For each method, sketch a graph showing

    1. The area represented by that approximation

    2. The area represented by the integral \(\int_{-1}^2 f(x) dx\)
    3. The area representing the error

  4. Based on your graphs in Question 3, determine whether each of the following are underestimates or overestimates. Explain your reasoning for each method.

    1. A left-hand sum

    2. A right-hand sum
    3. The midpoint method

    4. The trapezoid method

*Note: This is nonstandard nomenclature for Simpson’s method. What we call SIMP(n) is typically called SIMP(2n).