Fundamental Theorem of Calculus
- Author:
- Ken Schwartz
- Topic:
- Calculus, Integral Calculus
The Fundamental Theorem of Calculus Part I says . This can be a little difficult to navigate, with the in the limits of integration and the as the variable of integration. This app is intended to help make this concept more concrete using a real-life example.
On the left is a graph of a rate of change (derivative). It represents the rate , , at which a pump moves water. In this example, the pump starts pumping at a rate of 5 gallons per minute, but continuously slows down as it runs, dropping to 2 gallons per minute somewhere around 9 minutes later. Time for a new pump, I guess!
The amount of water that it pumps in a short time interval is found as the flow rate times . (Since is changing during , we would hold constant some value of within ). You can think of these as the height () and the width () of a rectangle on the left graph, and the amount pumped during as the area of this rectangle. So, the total amount pumped between times and is the sum of the areas of many of these narrow rectangles. As the number of rectangles increases, they get narrower, and the approximation of the area gets better. Thus we have a Riemann sum, .
On the right is plotted a graph of an accumulated quantity (integral). It represents the amount of water pumped at the rate since time . Let's fix at some point in time that we want to start measuring the amount of water pumped. Now, we let be the (arbitrary) point in time at which we want to know the total amount of water pumped since . Thus, the total amount of water pumped between times and is really a function , calculated by . This is what's graphed at the right - it's really .
So we've established the relationship between (the derivative) and (the integral). Now let's see how this relates to the Fundamental Theorem.
If you actually solve the "definite integral" , you would first find the antiderivative of (which is ), then plug in and and take the difference: . Notice that the app's default is so that what's graphed is just . You can use the slider to change the value of , which has the effect of shifting the graph vertically. Now let's take the derivative of . Since is a constant, its derivative is zero. And the derivative of is just . Thus, .
You can drag the "" and "" points on the -axis at the left to see the effects on . You can also type a new rate function for the graph at the left, and see the effects on the amount function on the right. (The software requires using "x" as the variable in the input box). Click the circular arrows at the top right of the left graph to reset the app.