Defining the Definite Integral as a Limit of Riemann Sums

Why the Definite Integral Needs a Limit

Suppose we want the exact signed area between a function and the on the interval . If the graph is curved, we cannot usually find the exact area by using one rectangle. Instead, we divide the interval into many smaller pieces and approximate the area with rectangles.

With a finite number of rectangles, we get an approximation. But what happens if we keep increasing the number of rectangles? The rectangles become narrower, the approximation becomes more accurate, and in the limit the approximation approaches the exact area.

This limiting process is the foundation of the definite integral.

The important idea is that the definite integral is not merely a convenient symbol for area. It is defined as the limit of these increasingly fine Riemann-sum approximations.

Step 1: Divide the Interval into Subintervals

Start with the interval . Divide it into equal subintervals.

The width of each subinterval is:

As becomes larger, becomes smaller. This is what allows the rectangles to fit the curve more closely.

Step 2: Choose a Sample Point

For each subinterval, choose a point at which to evaluate the function. A general sample point in the subinterval can be written as .

Different choices produce different types of Riemann sums:

• Left Riemann sum: use the left endpoint of each subinterval.
• Right Riemann sum: use the right endpoint.
• Midpoint Riemann sum: use the midpoint.
• General Riemann sum: choose any sample point inside each subinterval.

For the definition of the definite integral, the particular choice of sample points does not matter as the subintervals become arbitrarily small, provided the function is sufficiently well-behaved.

Step 3: Build the Riemann Sum

The approximate signed area of one rectangle is:

Adding all rectangles gives a Riemann sum:

This is still an approximation because there are only finitely many rectangles.

Step 4: Let the Number of Rectangles Approach Infinity

Now comes the key step. Let the number of subintervals increase without bound:

As , the width . The rectangles become infinitely thin in the limiting process, and the Riemann sum approaches the exact value of the area.

That limit is what we call the definite integral.

The Definition of the Definite Integral

For a continuous function on , the definite integral is defined by:

This formula is worth recognizing in several equivalent forms. The most common version uses right endpoints:

Here, because the right endpoint of the subinterval is .

Worked Examples

What the Limit Is Really Doing

It is easy to look at the notation and think that the limit is just a technical trick. It is more useful to understand what is happening geometrically.

• With rectangles, the approximation may be rough.
• With rectangles, the rectangles become narrower.
• As , the width of each rectangle approaches zero.
• The Riemann sum approaches one exact number.
• That exact limiting value is .

So the definite integral can be thought of as the result of taking the idea of "adding infinitely many infinitely thin pieces" and defining it precisely through a limit.

Riemann Sum Notation vs. Integral Notation

AP Calculus often asks you to move between these two forms. The correspondence is:

A useful way to read the Riemann sum is:

• tells you to add the contributions from all subintervals.
• is the function value at a sample point.
• is the width of each subinterval.
• tells you that the number of subintervals is becoming arbitrarily large.

Common Mistakes

Practice Problems