AP Calculus ABName______

5.2 Definite Integrals

  • I can express the area under a curve as a definite integral and as a limit of Riemann sums
  • I can compute the area under a curve using numerical integration procedures

In the previous section, we estimated distances and areas with finite sums, using LRAM, RRAM, and MRAM methods. In this section, we move beyond finite sums to see what happens in the limit as the terms become infinitely small and their numbers infinitely large.

In order to compute these sums, we will need to use Signma-notation for a sum, so let’s quickly review Signma-notation:

Riemann Sums

Georg Friedrich Bernhard Riemann (nothing like having a four-part name) lived from 1826-1866. The type of sum we are going to study are named after him because, while the limits of the sums we are about to learn about were used well before Riemann’s time, they were used without mathematical proof until Riemann was able to prove the existence of their limit in 1854. The proof of the existence of Riemann sums is beyond the scope of this course (that is one of my favorite phrases, because it means that we do not have to worry about the proof).

In the previous section, we used rectangles to approximate area under a curve. While all RAM sums are technically Riemann sums due to the way they are constructed, there are ways to construct Riemann sums so that the sums may be used for all types of functions and not just nonnegative functions.

The process to find a Riemann Sum is described below.

Look at on the closed interval

  • Divide into n subintervals by choosing points, say .
  • The intervals DO NOT need to be the same length. To keep notation consistent, let and
  • So the width of the 3rd interval is ; similarly, the length of the kth interval is
  • For the kth subinterval, its width is given by
  • In each subinterval , pick some point. It DOES NOT matter where in the interval you pick
  • On each subinterval, stand a rectangle that reaches from the x-axis to touch the curve at ; These rectangles can lie either above or below the x-axis
  • The area of each rectangle is and you will have n total rectangles
  • The sum of the area of all the rectangles will be

The sum , which depends on the partition P and the choice of numbers , is a

Riemann Sum for f on the interval [a, b]

As the partitions become finer and finer, we would expect the rectangles defined by the partitions to approximate the region between the x-axis and the graph of f with increasing accuracy.

The largest subinterval length of the partition P (remember, each partition can be a different length) is called the norm of the partition, and is denoted . As (or alternately as ), all Riemann Sums for a given function will converge to a common value.

The Definite Integral as a Limit of Riemann Sums

Let f be a function defined on a closed interval . For any partition Pof , let the numbers be chosen arbitrarily in the subintervals . If there exists a number I such that

no matter how P and the are chosen, then f is integrable on and I is the definite integralof f over .

Theorem:All continuous functions are integrable. That is, if f is continuous on , then its definite integral over exists

The Definite Integral of a Continuous Function on

Let f be continuous on and let be partitioned into n subintervals of equal length . Then the definite integral of f over is given by

Where each is chosen arbitrarily in the kth subinterval.

The definite integral is the “signed” area under a curve from a to b

Integral Notation

Example 1Example 2

Evaluate Evaluate

Example 3

Evaluate

More About Signed Area

Note that all the functions we have talked about so far have been nonnegative. However, we stated that the integral is the signed area under a curve. Therefore, if a function , then its area will be negative. Therefore,

or more commonly

If an integrable function has both positive and negative values on an interval , then

Exploration

It is a fact that .

With that information, determine the values of the following integrals.

1. 2.

3. 4.

5. 6.

7. 8.

Integrals on a Calculator

Not surprisingly, your calculator is able to handle numeric integrals. To calculate numeric integrals on your TI-Nspire CX, simply use the button and find the integral template. It looks like this:

Alternatively, you can also click menu 4:Calculus2:Numerical Integral to pull up the template.

Practice

Evaluate the following integrals on your calculator:

1. 2. 3.

Functions with Discontinuities

Earlier, we had a theorem that stated that all continuous functions are integrable. But what about discontinuous functions? As it turns out, some of them are integrable also.

Example 3

Find

Exploration 2

1.Explain why the function is not continuous on the interval . What kind of discontinuity occurs?

2.Show that

3.Show that