# The Fundamental Theorem of Calculus: What It Is and How It Works

There are some who say that the Fundamental Theorem of Calculus is one of the most important theorems in the history of mathematics. Here it is:

Break out the smelling salts!

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. To figure out the value of C for a particular value of s, just set the antiderivative equal to zero, plug the s value into x, and then solve for C.

Let’s determine the area function for f(t) = 10 when s = 0.

First, set the antiderivative equal to zero:

10x + C = 0

Then plug the s value into x and solve for C:

So for this particular area function, when s equals zero, C also equals zero.

(But note that C doesn't necessarily equal s. In fact, it usually doesn’t.)

Three area functions for f(t) = 10.

This figure shows why

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

You can also come up with this extra 20 with the method described earlier.

Set the antiderivative equal to zero:

10x + C = 0

Then plug the s value (-2) into x and solve for C:

And if you start sweeping out area at s = 3, the area function is

For the next example, look at the parabola

shown in the following figure.

You can easily compute the shaded area.

The area swept out from 0 to 3 is simply

If instead you want the area under the parabola between 2 and 3, there are two things you can do. First, you can determine the area function for s = 2.

Set the antiderivative equal to zero:

Then plug the s value of 2 into x and solve for C:

The area swept out between 2 and 3 is simply

The second method for figuring the area between 2 and 3 is to just compute

(the area between 0 and 3) and then take away the area you don’t want, namely

(the area between 0 and 2). Thus, the area between 2 and 3 is given by