Why Riemann Sums?

Posted on by

You are now ready to move into the study of integration, the other “half” of calculus. To integrate is defined as “to bring together or incorporate parts into a whole” (Dictionary.com).

The initial problem in integral calculus is to find the area of a region between the graph of a function and the x-axis with vertical sides. This is done by lining up very thin rectangles, finding their individual areas and incorporating them into a whole by adding their areas.

The way the rectangles areas are found and added is to use a Riemann sum. The width of each rectangle is a small distance along the x-axis and the length is the distance from the x-axis to the curve. As you use more rectangles over the same interval, their width decreases, and the approximation of the area becomes better.

Yes, that’s limits again. As the number of rectangles increases (\displaystyle n\to \infty ), their width decreases (\displaystyle \Delta x\to 0) and the (Riemann) sum approaches the area.

You will start by setting up some of these Riemann sums with a small number of rectangles to help you get the idea of what’s happening. (Lots of arithmetic here.)

Written in mathematical notation, a Riemann sum looks like this \displaystyle \sum\limits_{{n=1}}^{\infty }{{f\left( {{{x}_{n}}} \right)\Delta x}}. The interval on the x-axis is divided into subintervals of width \displaystyle \Delta x; these do not have to be the same, but almost always are. The \displaystyle f\left( {{{x}_{n}}} \right)  is the function’s value at some point, \displaystyle {{x}_{n}}, in each interval. So, \displaystyle f\left( {{{x}_{n}}} \right)\Delta x is the area of the rectangle for that subinterval. The sigma sign sums them up.

And the \displaystyle \underset{{\Delta x\to 0}}{\mathop{{\lim }}}\,f\left( {{{x}_{n}}} \right)\Delta x gives the area.

Most of the time the limit will not be easy to find, so you’ll avoid it! Soon you will learn a quick and efficient way to find the limits.

Riemann sums can be used in many other applications as you will soon learn.

Course and Exam Description Units 6. 1 and 6.2

1 thought on “Why Riemann Sums?”

  1. Hi Lin and AP Calculus colleagues –

    Your post states that the initial problem in integral calculus is finding the area of a region between a graph and the x-axis. This starting point is probably the most frequently used.

    For the approach I’ve found more effective, see my Dropbox posting,
    http://www.dropbox.com/sh/798k0tmbqnz4wvy/AADZMDbpCDsUNlWr0R2NDYL_a?dl=0
    for the PowerPoint and handout from my APAC session in Seattle last July.

    Students grasp the concept better if you start out with the problem, “How do you find a product of two numbers if one of the numbers varies? For instance, distance = (rate)(time), but what if the rate varies with time? For a constant rate, students easily see that the region under the rate-time graph is a rectangle, and thus (rate)(time) equals the area of that rectangle. If the rate varies, the distance still equals the area of the region, but now there is a reason for wanting to find that area. (For those who ask, “How can a distance equal an area?” I point out that both are numbers. The numbers can be the same for different physical quantities, like a dozen eggs and a dozen oranges.)

    First, I have students approximate the area by counting squares under an accurate graph. The next day I have them slice the region into trapezoids (not rectangles) to avoid the distraction of “negative areas.” Using a program to sum n trapezoid areas, they find that the sums have more decimal places stay the same as n increases. Thus the students get a “gut” feeing for limit of a sum.

    Riemann sums come later, before deriving the FTC, when it is more convenient to have just one sample point in each subinterval instead of two. The FTC becomes a source of empowerment. Students say, “Now I don’t to use those **** approximation methods all the time!” rather than, “Huh? What does an antiderivative have to do with volume of a solid of variable cross-sectional area?”

    Like

    Reply ↓

Leave a comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.