- Use sigma notation to evaluate sums of rectangular areas
- Find limits of upper and lower sums
- Use the limit definition of area to solve problems
In The Lesson The Calculus we introduced the area problem that we consider in integral calculus. The basic problem was this:
Suppose we are interested in finding the area between the axis and the curve of from to
We approximated the area by constructing four rectangles, with the height of each rectangle equal to the maximum value of the function in the sub-interval.
We then summed the areas of the rectangles as follows:
We call this the upper sum since it is based on taking the maximum value of the function within each sub-interval. We noted that as we used more rectangles, our area approximation became more accurate.
We would like to formalize this approach for both upper and lower sums. First we note that the lower sums of the area of the rectangles results in Our intuition tells us that the true area lies somewhere between these two sums, or and that we will get closer to it by using more and more rectangles in our approximation scheme.
In order to formalize the use of sums to compute areas, we will need some additional notation and terminology.
In The Lesson The Calculus we used a notation to indicate the upper sum when we increased our rectangles to and found that our approximation . The notation we used to enabled us to indicate the sum without the need to write out all of the individual terms. We will make use of this notation as we develop more formal definitions of the area under the curve.
Let’s be more precise with the notation. For example, the quantity was found by summing the areas of rectangles. We want to indicate this process, and we can do so by providing indices to the symbols used as follows:
The sigma symbol with these indices tells us how the rectangles are labeled and how many terms are in the sum.
Useful Summation Formulas
We can use the notation to indicate useful formulas that we will have occasion to use. For example, you may recall that the sum of the first integers is We can indicate this formula using sigma notation. The formula is given here along with two other formulas that will become useful to us.
We can show from associative, commutative, and distributive laws for real numbers that
Compute the following quantity using the summation formulas:
Another Look at Upper and Lower Sums
We are now ready to formalize our initial ideas about upper and lower sums.
Let be a bounded function in a closed interval and the partition of into subintervals.
We can then define the lower and upper sums, respectively, over partition , by
where is the minimum value of in the interval of length and is the maximum value of in the interval of length
The following example shows how we can use these to find the area.
Show that the upper and lower sums for the function from to approach the value
Let be a partition of equal sub intervals over We will show the result for the upper sums. By our definition we have
We note that each rectangle will have width and lengths as indicated:
We can re-write this result as:
We observe that as
We now are able to define the area under a curve as a limit.
Let be a continuous function on a closed interval Let be a partition of equal sub intervals over Then the area under the curve of is the limit of the upper and lower sums, that is
Use the limit definition of area to find the area under the function from to
If we partition the interval into equal sub-intervals, then each sub-interval will have length and height as varies from to So we have and
Since , we then have by substitution
as . Hence the area is
This example may also be solved with simple geometry. It is left to the reader to confirm that the two methods yield the same area.
- We used sigma notation to evaluate sums of rectangular areas.
- We found limits of upper and lower sums.
- We used the limit definition of area to solve problems.
In problems #1–2 , find the summations.
In problems #3–5, find and under the partition
In problems #6–8, find the area under the curve using the limit definition of area.
In problems #9–10, state whether the function is integrable in the given interval. Give a reason for your answer.
on the interval
on the interval
; (note that we have included areas under the axis as negative values.)
- Yes, since is continuous on
- No, since ;