#13: DEFINITE INTEGRAL

ANOTHER LIMIT PROBLEM

There was a different approach in using "small changes" than that which resulted in the derivative. This approach was dealing with sums. Look at the idea of speed and the distance traveled. If you are going at constant speed, then the distance traveled is just speed times time. This is a very simple formula that you probably used in grade school. But what if your speed is not constant? The formula does not apply. One approach is to divided up the time into very small units so that over that each the speed is "almost constant" over that unit. A representative speed could be then be determined by averaging the two speeds at the endpoints or by taking the speed at the midpoint. Over each of these units, the formula could be used and the distanced traveled in the unit could be found. The distances could then be added together to get the total distance. The smaller the units you used, the better your answer was. The unit size depends upon how many units you divide you time into. If you could let the number of units increase without bound, then your answer would be exact.

This was a very tedious chore but it worked. The exact answer to this was called the definite integral and is written as . Now you know why the integral symbol is an elongated "S". But an immediate question that comes to mind is "What is the connection between this limit of sums and the antiderivative, which used the same symbol?" The answer to that can be seen through the area problem.

AREA PROBLEM

Suppose you want to find the area, A(x), bounded by the function s(t) above from t = a to t = x, where x is some point between a and b. This function is exists but would take a lot of time to find using the above methods. You would approximate the area using rectangles (gray boxed below.) The area of each of these rectangles is the approximate height of the function, s(t), times the unit of time. Summing these up would give you the Area. In definite integral notation, this would be written as . Notice in this case the area represents the distance traveled from t = a to t = x.

Instead of actually finding the area, suppose you wanted to know what its derivative is. (Remember, mathematicians asks these kinds of questions which is how most of mathematics is developed.) To calculate the derivative, you must:

• First find A(x+h) - A(x). In the picture above, this is represented by the red rectangle.
• Next divide A(x+h) - A(x) by h. Since h is the base of the rectangle and the area of a rectangle is height times base, this quotient is just the height of the rectangle, s(t) for some value between x and x + h.
• Last you must take the limit as h approaches 0. Now as h approaches 0, the x + h approaches x. Since t is some value between x and x + h, this means that t approaches

Summarizing the above, this means that the derivative of A(x) is s(x), or that A(x) is the antiderivative of s(x)!

This gives you another way to find the distance traveled, which is the area under the curve.

• Let S(x) be an antiderivative of s(x). Then A(x) = S(x) + C.
• To evaluate C, observe that the distance traveled or area from t = a to t = a is 0.
  So, A(0) = 0 = S(a) + C. Solving for C gives C = - S(a).
• Therefore A(x) = S(x) - S(a)
• The area, or distance traveled, to t = b is A(b) = S(b) - S(a)

If the antiderivative can be found, this is a MUCH easier way of finding the definite integral. This is also the way you will calculate all definite integrals.

The connection between the definite integral and the antiderivative is called the Fundamental of Calculus.

To see visualizations of the sums for areas using rectangles and some examples on evaluating the definite integral visit the following web sites:

• Visual Calculus by Lawrence S. Husch at the University of Tennessee, Knoxville
• Areas. The two examples have a excellent demonstration on how smaller and smaller units approximate the area better.
• Fundamental Theorem of Calculus In the first link on this page, look at problem 2, 3 and 4. (Note, when the window opens for these problems, enlarge it so you can see the information at the bottom.) The third link verifies the Fundamental Theorem using a linear example. Skip the other links unless you are interested. Most of them involve proofs or trig functions.
• Areas by Murray Bourne, MSC, Ngee Ann Polytechnic, Singapore.

Note: Read section 5.6 to understand the summing process. It also tells you how calculators and computers calculate definite integrals. You will not be responsible for problems in this section.

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