Studying Mathematics Thread

[polymath257]

So what does a definite integral actually mean?

Say we have a continuous function f(x), with a domain interval [a,b]:

The definite integral of f(x) is the sum of all the values of f(x) within the domain [a,b]. Or, in other words, the net sum area of the region(s) bounded by the function graph and the x-axis.

As manually calculating such a sum is virtually impossible for most functions, we need to make use of the concept of the limit itself. Just like with derivatives, definite integrals can be defined in terms of the limit of some expression. But what expression should it be?

Well, let's approximate the total region area bounded by the graph (whether from the top or the bottom) using adjacent rectangles of equal widths and varying heights, each extending from the x-axis and having its top/bottom edge touching f(x) at some point, with the leftmost rectangle touching the x-axis at x=a with its bottom left corner and the rightmost rectangle touching the x-axis at x=b with its bottom right corner. We need equal widths in order to make easy use of the limit eventually.

Once the rectangles are drawn, we can approximate the area of interest by simply adding up the positive/negative areas of all the rectangles. This is an approximation of course, as there will be gaps in the region of interest that are not covered by the rectangles themselves and rectangle parts extending beyond the curve.

To get to the exact value of the area, you need to visualize the width of each rectangle equally shrinking at the same rate. As the rectangles become narrower and narrower, more equally wide rectangles are able to fit into the region, better approximating the area. As the width approaches 0, each rectangle in the region gradually evolves into a vertical straight line, with each line stopping at the graph at exactly one point. That's when you get the exact area value (since the height of each "fully shrunken rectangle" would exactly match the value of f(x) at the point it touches).

0 widths means infinite rectangles, and remember that the area of a rectangle is width * height. Therefore, the standard definition formula for the definite integral should now make sense.

As the number of rectangles approaches infinity, the sum of the net areas of the rectangles approaches the exact value of the total net area, or the definite integral.

For reference, here's the standard formula for the definite integral:
https://pasteboard.co/Hi0l1le.png

This is known as the Riemann integral. In it, we divide up the x-axis, the form the product of the length of the interval on the x-axis and the height of the function. Then we add up tttttt products. Finally we take a limit. All is well and good.

But there is another approach due to Lebesgue. Instead of dividing up the x-axis, divide up the y-axis. For each interval on the y-axis, look at the size of the set on the x-axis where the function values are in that interval (for nice functions, this is the sum of the lengths of a bunch of intervals). Again, multiply this size on the x-axis with the height on the y-axis, sum up, and take a limit.

The two methods give the same end result for 'nice' functions (read: continuous), but it turns out that the Lebesgue formulation works in far more generality and is much better when dealing with limit properties of functions. This is the start of the subject of 'Measure Theory'.