Integration

Integration is a method of calculation that calculates the area under a function.

When we integrate a function, we get another function:

\int f(x) = F(x)+C

For now, just use an Integration Chart to get from f(x) to F(x). Students typically memorize about half a dozen to a dozen of these.

If f(x)=5 then F(x)=5x+C

For our purposes here, C=0. Elsewhere we will explain its value.

If f(x)=100 then F(x)=100x+C

Let a be any value:

If f(x)=a then F(x)=ax+C

We looked at two examples and then we took a guess at the general formula. Now, can we make sense of it?

We have a trick to process F(x).

F(x) \Big|_a^b = F(b) - F(a)

Integrating 1 gives us x, so the following is true:

\int 1 dx = x + C

We have written the “1” even though it is OK to omit it, so you can see that there is a function to the left of the “dx”.

When we integrate from 1 to 3, we write it as follows:

\int_1^3 1 dx = x \big|_1^3 = (3+C)-(1+C) = 2

Do you see that, because of the subtraction, the C constants cancel?

Another example is shown, this time f(x)=x. Assume we want the area from x=2 to x=6.

\int_2^6 x \: dx = \big |_2^6 \dfrac {1}{2} x^2 = \dfrac {1}{2}36 - \dfrac{1}{2}4 = 16

If you draw the function y=x on a graph, and then shade the area under the function from x=2 to x=6, can you see that you get the area by starting with half the area of a 6×6 box and then subtracting half the area of a 2×2 box?