Math& 152
Riemann Sums and the Definite Integral
Name:
Labs are meant to be completed partially on your own and partially in your group of 2 – 3
people. The portion that is called “Before the Lab” is to be done by each member of the
group. The portion entitled “In the Lab” will also be completed by each member of the
group, however the group will only turn in one electronic copy of the lab.
GOALS:

To approximate the area under a curve by summing the areas of coordinate
rectangles.

To develop this idea of Riemann sums into a definition of the definite integral.

To understand the relationship between the area under a curve and the definite
integral.
BEFORE THE LAB
1. The figure below shows the graph of a function f on the interval [a, b]. We want to write an
expression for the sum of the areas of the four rectangles that will depend only upon the
function f and the interval endpoints a and b. The four subintervals that form the bases of the
rectangles along the x-axis all have the same length; express the length of each base in terms
of a and b. How many subinterval lengths is x2 away from a = x0? Write expressions for x1,
x2, x3, and x4 in terms of a and b. What are the heights of the four rectangles? Multiply the
heights by the lengths, add the four terms, and call the sum R(4).
2.
Generalize your work in part (a) to obtain an expression for R(n), the sum of
rectangular areas when the interval [a, b] is partitioned into n subintervals of equal length
and the right-hand endpoint of each subinterval is used to determine the height of the
rectangle above it. Write your expression for R(n) using summation notation. In order to do
this, first figure out a formula for xk, the right-hand endpoint of the kth subinterval. Then
check that your formula for xk yields the value b when k takes on the value n.
3.
Let L(n) denote the sum of rectangular areas when left-hand endpoints rather than
right-hand endpoints are used to determine the heights of the rectangles. Add some details
to the figure on the front page to illustrate the areas being summed for L(4). What
modification in the expression for R(n) do you need to make to get a formula for L(n)?
4.
Have the instructor check your formulas for R(n) and L(n).
___________________
Math& 152
Riemann Sums and the Definite Integral
Names:
IN THE LAB
1. Consider the function 𝑓(𝑥) = 3𝑥 on the interval [1, 5].
a. Apply your formulas for R(4) and L(4) to this function. On a piece of paper (which you
will hand in) sketch a picture to illustrate this situation. Use geometry to determine the
exact area A between the graph of f and the x-axis from x = 1 to x = 5.
b. Now use the computer to get values for R(n) and L(n) for n = 40. How are the sizes of
R(40) and L(40) related to the area A? Explain this relationship and make a conjecture (an
educated guess) about what should happen for larger and larger values of n. Test your
conjecture with a few more values of n of your own choosing and record your results in a
table.
2. Consider the function f(x) = 1/√𝑥 on the interval [0.1,10]. Plot the graph of f and use
the ideas developed above to approximate the area A under the graph of f and above the
given interval. What is the relation among R(n), L(n) and the area A now? Explain any
difference that you see between the situation here and the situation in problem (1).
3. Now let f(x) = 4-x2 on the interval [-2, 2]. Plot the graph and estimate the area under it
on this interval. How are the values of R(n) and L(n) related this time? What has changed
and why? Does this prevent you from getting a good estimate of the area? Explain your
reasoning.
4. So far we have worked with functions that are nonnegative on their domains. There is,
however, nothing in the formulas you have developed that depends on this. Here we
consider what happens geometrically when the function takes on negative values.
a. consider the function f(x) = 3x on the interval [-4, 2]. On a separate piece of paper
(which you will turn in) sketch the graph of f on this interval and include appropriate
rectangles for computing R(3) and L(3). On subintervals where the function is negative,
how are the areas of the rectangles combined in obtaining the overall value for R(3) or L(3)?
What value do R(n) and L(n) seem to approach as n increases? How can you compute this
value geometrically from your sketch?
b. Now consider the function f(x) = ex^2 - 2x - 14 on the interval [2, 3]. Use your
computer to plot it. Determine the value that R(n) and L(n) seem to approach. Explain with
the help of your graph why you think this is happening.
Here is a summary of the two important points so far:


Approximation by rectangles gives a way to find the area under the graph of a
function when that function is nonnegative over the given interval.
When the function takes on negative values, what is approximated is not an actual
area under a curve, though it can be interpreted as sums and differences of such
areas.
In either case the quantity that is approximated is of major importance in calculus and in
mathematics. We call it the definite integral of f over the interval [a, b]. We denote it by .
5. For each of the definite integrals below, use either left or right-hand Riemann sums to
approximate its value. Make your own decision about what values of n to use. Then use the
definite integral command either on Maple (use the definite integral on the left-hand side
under the Expression tab) or on your calculator (on the TI this is under MATH and is called
fnInt)to obtain another approximation. Comment on the degree to which the values agree.
1
a. ∫0 𝑒 𝑥 𝑑𝑥
3
b. ∫1 (𝑥 3 − 3𝑥 2 − 2𝑥 + 3)𝑑𝑥
3
c. ∫−1 𝑠𝑖𝑛(𝑥 2 )𝑑𝑥