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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:

image0.png

Break out the smelling salts!

image1.png

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:

image2.png

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

image3.png

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

Three area functions for <i>f</i>(<i>t</i>) = 10.
Three area functions for f(t) = 10.

This figure shows why

image5.png

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

image6.png

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

image7.png

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:

image8.png

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

image9.png

For the next example, look at the parabola

image10.png

shown in the following figure.

image11.jpg

You can easily compute the shaded area.

image12.png

The area swept out from 0 to 3 is simply

image13.png

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:

image14.png

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

image15.png

The area swept out between 2 and 3 is simply

image16.png

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

image17.png

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

image18.png

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

image19.png
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