Calculus 2 Labs
Lab 2: Integral approximations  part 1
Lab Preparation: Answer the following questions individually and bring your writeup to class.

Under what conditions is the lefthand sum for a definite integral an underestimate? An overestimate? Why?:

Under what conditions is the righthand sum for a definite integral an underestimate? An overestimate? Why?

The average of the lefthand sum and the righthand 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 oneinch 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.

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

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

A lefthand sum, LEFT(4)

A righthand sum, RIGHT(4)

The midpoint method, MID(4)

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

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


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

The area represented by that approximation
 The area represented by the integral \(\int_{1}^2 f(x) dx\)

The area representing the error


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

A lefthand sum
 A righthand sum

The midpoint method

The trapezoid method

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