How Statistics Shows the Connection Between Differentiation and Integration
Still trying to grasp how differentiation and integration work? No problem: you can use statistics to help you. By studying the relationship between two simple graphs, you’ll understand the relationship between differentiation and integration (and, what’s more, you don’t need to know any statistics at all to understand this idea!).
The graphs in question are a frequency distribution graph and a cumulative frequency distribution graph (you may have run across such graphs in a newspaper or magazine). Take a look at the figure.
The upper graph in the figure shows a frequency distribution histogram of the annual profits of Widgets-R-Us from January 1, 2001 through December 31, 2013. The rectangle marked ’07, for example, shows that the company’s profit for 2007 was $2,000,000 (their best year during the period 2001–2013).
The lower graph in the figure is a cumulative frequency distribution histogram for the same data used for the upper graph. The difference is simply that in the cumulative graph, the height of each column shows the total profits earned since 1/1/2001. Look at the ’02 column in the lower graph and the ’01 and ’02 rectangles in the upper graph, for example. You can see that the ’02 column shows the ’02 rectangle sitting on top of the ’01 rectangle which gives that ’02 column a height equal to the total of the profits from ’01 and ’02. Got it? As you go to the right on the cumulative graph, the height of each successive column simply grows by the amount of profits earned in the corresponding single year shown in the upper graph.
Okay. So here’s the calculus connection. Look at the top rectangle of the ’08 column on the cumulative graph (let’s call that graph C for short). At that point on C, you run across 1 year and rise up $1,250,000, the ’08 profit you see on the frequency distribution graph (F for short). Slope = rise/run, so, since the run equals 1, the slope equals 1,250,000/1, or just 1,250,000, which is, of course, the same as the rise. Thus, the slope on C (at ’08 or any other year) can be read as a height on F for the corresponding year. (Make sure you see how this works.) Since the heights (or function values) on F are the slopes of C, F is the derivative of C. In short, F, the derivative, tells you about the slope of C.
The next idea is that since F is the derivative of C, C, by definition, is the antiderivative of F (for example, C might equal 5x3 and F would equal 15x2). Now, what does C, the antiderivative of F, tell you about F? Imagine dragging a vertical line from left to right over F. As you sweep over the rectangles on F — year by year — the total profit you’re sweeping over is shown climbing up along C.
Look at the ’01 through ’08 rectangles on F. You can see those same rectangles climbing up stair-step fashion along C (see the rectangles labeled A, B, C, etc. on both graphs). The heights of the rectangles from F keep adding up on C as you climb up the stair-step shape. And you’ve seen how the same ’01 through ’08 rectangles that lie along the stair-step top of C can also be seen in a vertical stack at year ’08 on C. The cumulative graph is drawn this way so it’s even more obvious how the heights of the rectangles add up. (Note: Most cumulative histograms are not drawn this way.)
Each rectangle on F has a base of 1 year, so, since
the area of each rectangle equals its height. So, as you stack up rectangles on C, you’re adding up the areas of those rectangles from F. For example, the height of the ’01 through ’08 stack of rectangles on C ($8.5 million) equals the total area of the ’01 through ’08 rectangles on F. And, therefore, the heights or function values of C — which is the antiderivative of F — give you the area under the top edge of F. That’s how integration works.
Okay, you’re just about done. Now let’s go through how these two graphs explain the relationship between differentiation and integration. Look at the ’06 through ’12 rectangles on F (with the bold border). You can see those same rectangles in the bold portion of the ’12 column of C. The height of that bold stack, which shows the total profits made during those 7 years, $7.75 million, equals the total area of the 7 rectangles in F. And to get the height of that stack on C, you simply subtract the height of the stack’s bottom edge from the height of its upper edge. That’s really all the shortcut version of the fundamental theorem says: The area under any portion of a function (like F) is given by the change in height on the function’s antiderivative (like C).
In a nutshell (keep looking at those rectangles with the bold border in both graphs), the slopes of the rectangles on C appear as heights on F. That’s differentiation. Reversing direction, you see integration: the change in heights on C shows the area under F. Voilà: differentiation and integration are two sides of the same coin.