# The Fundamental Theorem of Calculus

The fundamental theorem of calculus is one of the most important theorems in the history of mathematics. It states that, given an area function Af that sweeps out area under f (t),

the rate at which area is being swept out is equal to the height of the original function. So, because the rate is the derivative, the derivative of the area function equals the original function:

Because

you can also write the above equation as follows:

Break out the smelling salts.

Now, because the derivative of Af (x) is f (x), Af (x) is by definition an antiderivative of f (x). Check out how this works by looking at a simple function, f (t) = 10, and its area function,

According to the fundamental theorem,

Thus Af must be an antiderivative of 10; in other words, Af is a function whose derivative is 10. Because any function of the form 10x + C, where C is a number, has a derivative of 10, the antiderivative of 10 is 10x + C. The particular number C depends on your choice of s, the point where you start sweeping out area. For a particular choice of s, the area function will be the one function (out of all the functions in the family of curves 10x + C) that crosses the x-axis at s. To figure out C, set the antiderivative equal to zero, plug the value of s into x, and solve for C.

For this function with an antiderivative of 10x + C, if you start sweeping out area at, say,

or just 10x. (Note that C does not necessarily equal s. In fact, it usually doesn’t

When s = 0, C often also equals 0, but not for all functions.)

The figure shows why Af (x) = 10x is the correct area function if you start sweeping out area at zero.

In the top graph in the figure, the area under the curve from 0 to 3 is 30, and that’s given by

And you can see that the area from 0 to 5 is 50, which agrees with the fact that

If instead you start sweeping out area at s = –2 and define a new area function,

so C equals 20 and Bf (x) is thus 10x + 20. This area function is 20 more than Af (x), which starts at s = 0, because if you start at s = –2, you’ve already swept out an area of 20 by the time you get to zero. The figure shows why Bf (3) is 20 more than Af (3).

And if you start sweeping out area at

and the area function is

This function is 30 less than Af (x) because with Cf (x), you lose the 3-by-10 rectangle between 0 and 3 that Af (x) has (see the bottom graph in the figure).

An area function is an antiderivative. The area swept out under the horizontal line f (t) = 10 from some number s to x, is given by an antiderivative of 10, namely 10x + C, where the value of C depends on where you start sweeping out area.

Now take a look at some graphs of Af (x), Bf (x), and Cf (x).

(Note that the previous figure doesn’t show the graphs of Af (x), Bf (x), and Cf (x). You see three graphs of the horizontal line function, f (t) = 10; and you see the areas swept out under f (t) by Af (x), Bf (x), and Cf (x), but you don’t actually see the graphs of these three area functions.)

The second figure shows the graphs of the equations of Af (x), Bf (x), and Cf (x) which you worked out before: Af (x) = 10x, Bf (x) = 10x + 20, and Cf (x) = 10x – 30. (As you can see, all three are simple, y = mx + b lines.) The y-values of these three functions give you the areas swept out under f (t) = 10 that you see in the first figure. Note that the three x-intercepts you see in the second figure are the three x-values in the first figure where sweeping out area begins.

You have already worked out that Af (3) = 30 and that Af (5) = 50. You can see those areas of 30 and 50 in the top graph of the first figure. In the second figure, you see these results on Af at the points (3, 30) and (5, 50). You also saw in the first figure that Bf (3) was 20 more than Af (3); you see that result in the second figure where (3, 50) on Bf is 20 higher than (3, 30) on Af . Finally, you saw in the first figure that Cf (x) is 30 less than Af (x). The second figure shows that in a different way: at any x-value, the Cf line is 30 units below the Af line.

A few observations. You already know from the fundamental theorem that

(and the same for Bf (x) and Cf (x)). That was explained previously in terms of rates: For Af, Bf, and Cf, the rate of area being swept out under f (t) = 10 equals 10. The second figure also shows that

(and the same for Bf, and Cf), but here you see the derivative as a slope. The slopes, of course, of all three lines equal 10. Finally, note that the three lines in the second figure differ from each other only by a vertical translation. These three lines (and the infinity of all other vertically translated lines) are all members of the class of functions, 10x + C, the family of antiderivatives of f (x) = 10.