INTEGRATION

INTEGRATION

1. Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the area between a function and the x-axis like this

2. Indefinite Integral

3. Slices

   • We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate):
   • We can make Δx a lot smaller and add up many small slices (answer is getting better):
   • And as the slices approach zero in width, the answer approaches the true answer. We now write dx to mean the Δx slices are approaching zero in width.

4. Example: 2x

5. The symbol for "Integral" is a stylish "S" (for "Sum", the idea of summing slices):

   • After the Integral Symbol we put the function we want to find the integral of (called the Integrand) and then finish with dx to mean the slices go in the x direction (and approach zero in width).
     • And here is how we write the answer:

6. It is the "Constant of Integration". It is there because of all the functions whose derivative is 2x:

7. How do we integrate other functions?

8. A Definite Integral has actual values to calculate between (they are put at the bottom and top of the"S"):

   • We find the Definite Integral by calculating the Indefinite Integral at a, and at b, then subtracting:

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