The Cornerstone of Integration
Alright, listen tight. Riemann sums are probably not something that pop up in tests a lot, but they are extremely helpful for a complete understanding of integration. Also, take note that Riemann sums inspired integration. Suppose you wish to inspire something as important as integration in the future. You might want to read carefully now.
Who the hell is Riemann, anyways?
Georg Friedrich Riemann is a German mathematician who made great contributions to mathematics. He was born in 1826 in Germany. He was the founder of a section of geometry, called Riemannian geometry. He died in Italy at the early age of 39, due to Tuberculosis.
Some 200 or so years ago, a difficult problem emerged. How does one calculate the exact area under a curve?
Riemann sums provide a way to solve the reputed "unsolvable problem"-- splitting the area we wish to find into little, little pieces. How exactly we split it is not really a concern, because the sum of the split parts is supposedly equal to the area in question.
There are many ways to split the area. we can cut it horizontally, we can cut it vertically, or I can just carve the area up into random pieces. The idea is to split it into pieces that are easy to calculate, so my way is probably out, but...
Usually, there are three types of Riemann sums. the left Riemann sum, the middle Riemann sum, and the left Riemann sum.
When you think of the Riemann sum, think of histograms (or bars). Because this is what Riemann sums would look like, if we put them in a graph.
The key idea is to replace the area under the function (the one that has a smooth edge) with an edgy histogram, or bar graph, that approximates the area. Calculate the areas of the bars and then sum them up. The sum is going to be an approximation, not the exact value, as long as the bars have a certain width.
This is to say, if they are infinitely thin, their sum is going to be the exact value of the area. Which is true, when one thinks of it.
The graph above gives you an idea what it's all about. the upper left one, the one with intersections on the right side of the bar, is called the right Riemann sum. and the one that had intersections in the middle of the bar, is called the middle Riemann sum. I won't even bother to tell you what is a left Riemann sum.
And as you can see in the graphs, the thinner the bars are, the better an approximation it is.
Hopefully, by now, you have read the 'leading thought' section. Now you are prepared for the precise definition of Riemann sums.
The definition is something that looks like this:
Now, what does this thing mean? Going over this bulk tells us rather slowly that this is just an operation that goes from one side of the graph to the other, gradually draw a histogram that approximates the area beneath the curve.
Or what I have been trying so hard to say in the leading thought part.
Now, remember that we have learned about limits, and as calculus is the study of limits, we expect Riemann sums to connect in someway to limits. Here is how (for now, you can ignore the sign on the left):
What the above expression means is just a simple operation. consider the bar graph we used to approximate the function. Of course, each bar has a width. What if the width is zero? Than the sum of the areas of those "infinitely thin" bars would be equal to the exact area under the curve.
The problem remains how to implement this criterion (zero width). The expression above provides an excellent solution.
Consider appropriating the area with infinitely many bars. the width of each bar would be something (the width of the area beneath the curve) divided by infinity, which is zero. That is because any constant divided by infinity is zero. So, this is equal to approximating the area with a collection of infinitely many lines beneath the curve.
And the length of a line has a mathematical significance: it is the value of the function at that point. So the sum of all the lengths would be equal to adding up every value at every single point.
And with that, mission accomplished. All that is left to do is to evaluate the limit.
Such a calculation is often messy. That's why you need to look at the next section, the Fundamental Theorems of Calculus.