Chapter 5 Section 2 Riemann Sums

In this section you will learn how to represent the sum of the area of the rectangles used to estimate the area under a curve using mathematical notation. The first section of this chapter introduced to you the methods used to estimate the area under the curve, now you will see how we represent these methods in the form of an equation.

Let’s begin with a very general equation, and progress from there. When we estimate the area under the curve we are adding the area of several rectangles whose heights are close to the y-values of the graph.

Let A be a function for the area under a curve and R be a rectangle under the curve.

Thus A = R₁ + R₂ + R₃ + R₄ + R₅ + R₆ …+ R_n

How can we make this equation more specific? By representing the area of the rectangles, not simply by R, but with an expression that defines the area of a rectangle. So then, what is the area of a rectangle? That’s easy, base times height. But what is the base and the height of a rectangle under a curve. Let’s look at a graph to answer that question.

First of all, let’s determine which of the three methods we discussed in the previous section. This graph uses the right end point method. That means that the height in our b*h formula that height will be the y-coordinate corresponding with the top right point of the rectangle. So the height of the first rectangle is f (X₂). What is the base? X₂ – X₁.

Let’s assume that the width of each triangle is the same, and call “X₂ – X₁” “∆X”. Then, if we use the right end point method, the formula for the area under the graph is as follows:

A= f (X₂)* ∆X + f (X₂₊₁)* ∆X … + f (X_2+n)* ∆X

There are only minor changes in the formula if we use the left point or midpoint method. For instance, the left point formula would be:

A= f (X₁)* ∆X + f (X₁₊₁)* ∆X … + f (X_1+n)* ∆X

Here’s a good short cut for calculating width of your rectangles. Let’s say that in a problem from the book, you are asked to estimate the area under a curve on the interval [a, b], and you must use n subintervals (or rectangles—in problems, they will often be referred to as subintervals) of equal width. The formula to use to calculate the width of each subinterval is as below:

(b – a) / (n)

Thus ∆X = (b – a) / (n)

So far we’ve been looking at fairly inaccurate estimations of the area under a curve. You can see that from the graph I used above. What might make our estimations more accurate? Using more rectangles would—in fact the most accurate estimation of the area under a curve would have infinite subintervals. Let’s express that in an equation.

A = Lim_{n→ ∞} { f (X₂)* ∆X + f (X₂₊₁)* ∆X … + f (X_2+n)* ∆X}

(Notice that this equation represents subintervals using right end point method)

Can we simplify this equation anymore? Yes, see the below formula:

A = Lim_{n→ ∞} {*∆X}

The sigma stands for a sum—it tells you to add f(x₁) + f(x₂) +… f(x_n). You should recognize this symbol from the sequence unit of Pre-Calc. You might also remember this rule from sigma notation:

=

Where c is a constant.

Do we have any constants in the formula for area? Yes, ∆X will always be the same, as long as the width of each rectangle is the same, and for the most part this will be the case. Now our equation look like this

A = Lim_{n→ ∞} ∆X {}

I’m guessing most of you forgot the rules and short cuts that help you solve sigma equations. Here’s a list of various sigma equations that you should memorize.

You will use these short cuts to solve some to these problems, so keep them in mind. Look at the example problems to see how we can use these rules and short cuts.

The last thing that we need to go over are Riemann Sums. That’s just a fancy name for everything we have just gone over on this page. In Riemann Sum problems they will give you a graph or a data table and will tell you the number of subintervals they want you to use, and often whether these intervals need to have equal width and you must estimate the area under the graph. It is fairly simple, the most difficult part is deciding what each rectangle’s width must be, and thus the sample points you will use. There will probably be a Riemann Sum on the AP free response section.