The purpose of this set of notes is to relate two very important topics in integral calculus together. The first topic is the concept of a Riemann sum and the second topic is the definite integral. Let us start with the Riemann sum.

To understand this topic, lets us talk about a numerical method that is based on this topic. The method that I am talking about is the Rectangular Approximation Method. This method is based on dividing the area under a curve into a finite number of rectangles, calculating the area of each rectangle, then summing up the areas to approximate the area under the curve. Let us look at an example.

EXAMPLE 1:

Determine the area under the curve and above the x-axis for the function f (x) = -x ² + 4 on the interval [0, 2].

SOLUTION: Consider the function f (x) = -x ² + 4 on the interval [0, 2], and its graph. To do the rectangular approximation method, we need to know how many subintervals we are going to be using. I like to pick n to be an even number, so let n = 8. Now we will figure the length of each subinterval, D x.

f (x) = -x ² + 4

The next step is to construct two tables. One table will list the right and left endpoints and the other the midpoint of the interval. I will construct three different approximations. The first approximation, called the right rectangular approximation method, uses the right endpoint of the interval to calculate the height of the rectangle. The second approximation, called the left rectangular approximation method, uses the left endpoint of the interval to calculate the height of the rectangle, and the last approximation, the midpoint rectangular approximation method, uses the midpoint of the interval to calculate the height of the rectangle.

LEFT/RIGHT ENDPOINT TABLE

MIDPOINT TABLE

x ₀

0

4

x _a

.125

3.984275

x ₁

.25

3.9375

x _b

.375

3.859375

x ₂

.5

3.75

x _c

.625

3.609375

x ₃

.75

3.4375

x _d

.875

3.234375

x ₄

1

3

x _e

1.125

2.734375

x ₅

1.25

2.4375

x _f

1.375

2.109375

x ₆

1.5

1.75

x _g

1.625

1.359375

x ₇

1.75

.9375

x _h

1.875

.484375

x ₈

2

0

The Left Endpoint Rectangular Approximation (LRAM) starts with x ₀ and ends with x ₇. Here is the LRAM ₈ for this function.

The Right Endpoint Rectangular Approximation (RRAM) starts with x ₁ and ends with x ₈. Here is the RRAM ₈ for this function.

The Midpoint Rectangular Approximation (MRAM) uses the midpoints of the intervals. The values for the midpoints are under the section of the table labeled MIDPOINT TABLE. Here is the MRAM ₈ for this function.

Notice that the MRAM value is in between the LRAM and the RRAM value. The reason for this is that the height of the rectangle is taken from the middle of the interval and not at the endpoints. This allows for more coverage of the area under the curve.

Also notice that if I take a finer partition of the interval, the approximation will get closer to the true value of the definite integral of the function over the closed interval.

Now, let us divide the interval [0, 2] into n subintervals.

Using the following formulas from algebra, we will simplify the above sum.

Now take the limit of RRAM _n f (x) as n ® ¥ .

EXAMPLE 2:

Determine the area under the curve and above the x-axis for the function f (x) = -x ² + 4 on the interval [-1, 0].

SOLUTION: Consider the function f (x) = -x ² + 4 on the interval [-1, 0]. Let us divide this closed interval into n subintervals.

 Now, evaluate the limit of RRAM _n f (x) as n ® ¥ .

Notice that I have been using the right endpoint rectangular approximation. The reason why I prefer to use the right endpoint is the fact that you end with the right endpoint. It is a personal preference. The other two methods work just as easily.

EXAMPLE 3:

Determine the area under the curve and above the x-axis for the function f (x) = x ³ - x on the interval [0, 2].

SOLUTION: Consider the function f (x) = x ³ - x on the interval [0, 2] with n subintervals. Let us find the RRAM _n f (x).

 Again, let us evaluate the limit of RRAM _n f (x) as n ® ¥ .

Now that we have talked about the rectangular approximation method, let us talk about how this relates to the Riemann sum. Notice that in example 1 I mentioned the word integral. The rectangular approximation method is one of many numerical methods for calculating the definite integral. In fact, the rectangular approximation method is considered a type of Riemann sum. From the concept of Riemann sum the concept of a definite integral discussed.

DEFINITION:

Let f (x) be a function defined on a closed interval [a, b].

EXAMPLE 4:

Express the following limit as a definite integral over the given interval.

SOLUTION:

EXAMPLE 5:

Express the following limit as a definite integral over the given interval.

SOLUTION:

The second part of the Fundamental Theorem of Calculus states that if f is continuous at every point of [a, b] and F is any antiderivative of f on [a, b], then

EXAMPLE 6:

SOLUTION:

This is the same answer that is in example 2.

EXAMPLE 7:

SOLUTION:

EXAMPLE 8:

SOLUTION:

EXAMPLE 9:

SOLUTION:

 Before we go any farther, let us discuss some properties of the definite integral.

1.

2.

3.

4.

5.

6.

If max f and min f are the maximum and minimum values of f on [a, b], then

7.

EXAMPLE 10:

Suppose that f and g are continuous and that

Find

DEFINITION:

If f is integrable on [a, b], its average value on [a, b] is

FACT:

If f is continuous on [a, b], then at some point c in [a, b],

The above theorem is an extension of the above definition, and I will now do an example using it.

EXAMPLE 11:

Find the average value of f (x) = 3x ² - 3 on [0, 1]. At what points in the given interval does f (x) assume the average value?