The Fundamental Theorem of Calculus

Calculus has given us two powerful techniques. Integrals let us find the area under a graph, and derivatives find its slope, or rate of change. You might wonder if it is possible to combine these two concepts, and find the rate of change of the area under a graph.

The method of doing this is almost identical to finding the derivative. The secret to finding derivatives was to find the rate of change of two points, and let those points get closer and closer together until they were, more or less, identical. To find the rate of change of the integral we will find the area under the region defined by two points on the graph. Then let those points get closer and closer together until they are, more or less, the same point.

Look at the above graph of a generic function. Think of what happens if we add the interval from x to x + h. As h gets smaller and small the interval we add gets skinnier and skinnier. When it becomes infinitely thin the area we add is simply equal to the height of the graph. And that value is f(x). The derivative of an integral is the original function!

Lets see if we can use this insight to find the integral. Suppose we take the function f(x) = x². We now know that the derivative of the integral is x². So what function has a derivative of x²? Try and guess (really, grab a pencil and write it down).

The answer is ¹⁄₃x³. Apply the power rule and you’ll see that it works out.

If we apply differentiation in reverse we find the following.

Let f(x) = axⁿ
F(x) = integral of f(x) = ¹⁄_n+1axⁿ⁺¹

The Fundamental theorem of calculus: the derivative of the integral of f(x) is f(x). Similarly, the integral of the derivative of f(x) is also f(x).

The fundamental theorem of calculus says that we’ll never need to take a Riemann sum again! Instead well simply reverse the process of differentiation. This is called the anti-derivative.

Applications

Lets apply this process to the problem we did with Riemann sums. One of them was finding the area under f(x)=x² from 0 to 10.

The integral F(x) = ¹⁄₃x³.

Area from 0 to 10 = F(10) – F(0) = 1000/3 – 0 = 333.333

That is the exact answer we got before. And the formula clears up a minor mystery, which is why it seemed suspicious that the answer was exactly one-third of 1000.

There is another detail that may seem mysterious, which is taking f(10) – f(0). But F(x) gives the area under the graph throughout its entire graph up to point x. f(x) could be thought of as taking the area under the graph at precisely the point x. But we don’t want the area under the entire graph, we only want the area from x = 0 to x = 10. So you subtract one value from the other to get the region in which you are interested.

Now that we have the fundamental theorem of calculus, we can find the integrals of all types of functions, such as f(x) = x⁴⁸⁴. That is a heck of a lot easier than Riemann sums!