What is the relationship between definite integral and limit?

Answer 1

A definite integral can be defined as the limit as the norm of the partition approaches zero of a function value times a difference in its independent variable. Since I do not have access to a proper symbol library on here, I will try to write it out as best as I can:

lim as p->0 of sum from i=1 to n of f(xi)*(delta-x).

You may know about Riemann sums, which, in functions of one variable, approximate an integral using rectangular partitions of the area under a function's curve that go from the axis in question to some defined value on the curve in height and are some defined width of independent variable.

For instance, to approximate the area between the function y=4-x2 and the x-axis, the following could be done:

• The function has zeroes at x=-2 and x=2, so the total length of x to be partitioned is 4 units.
• You can choose any way to divide up this 4-unit length, but the most common way is to pick even intervals based on a desired number of rectangles. If you were asked to approximate this area with four rectangles, each partition would be of width 1 unit, since 4/4=1. The use of more rectangles causes the approximation to get more accurate. In truth you do not need to use equal partitions, but for most math classes equal partitions are asked for.
• Using the above 4-rectangle approach, you now have four equal intervals, [-2,-1], [-1,0], [0,1], [1,2]. You can choose any x-value on these intervals to plug into the function to get its corresponding y-value that will serve as the "height" of each rectangle. Generally if you choose the left-most value of one interval, you choose the left-most value of all intervals, but once again this is not necessarily the only way to do it, simply the most easily approachable.
• By constructing four rectangles, all of width 1 and of height y=4-x2 where x is some x on each interval, you get four rectangles whose areas can be added together to approximate the definite integral of y=4-x2 from -2 to 2 with respect to x.

Now, that was kind of a long-winded review, but I want to make sure you know that before I continue, and I can refer back to that to simplify my explanation of how limits relate to integrals. As I said above, as the number of partitions you use increases, the approximation of the integral gets more and more accurate. It follows a behavior similar to a limit in that it approaches closer and closer to the actual value of the integral. But, as you increase the number of partitions, the width of the partitions gets smaller and smaller since you are fitting more rectangles into the same space. To get the true value of the integral, you would need an infinite number of zero-width rectangles. This is impossible to definitively calculate, but by taking the limit as the width of the partitions approaches zero, we can find the exact value of the integral.