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Lab 2: Approximating definite integrals - 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 Instructions: 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 \(\displaystyle\int_{-1}^2 f(x) dx\).

  2. Find the approximations for \(\displaystyle\int_{-1}^2 f(x) dx\) 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 numerical value of 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 \(\displaystyle\int_{-1}^2 f(x) dx\)

    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. Don’t merely say you have an overestimate because “the rectangles are above the function.” Instead say what characteristics of the function cause the “rectangles” to produce overestimates.

    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).