12 - Algebra of Definite Integration

We learned about the indefinite integrals and practiced rules of integration.

However, definite integration has a very simple rule:

The definite integral between two points a and b of a function f(x) is equivalent to the difference between the primitive at the endpoints.

To clarify:

Suppose that you have some function , integrable, where is its primitive:
If we wanted the definite integral between a and b, we would have:

Because of the way integrals are, we can "split" a definite integral into a sum of definite integrals:
Think of it as connecting two pieces of rope!

The negative of a definite integral

If you have a negative of a definite integral it's the same as just swapping the upper and lower bounds:

You don't add the c to definite integrals!

The Integral Mean Value Theorem

Much like the Lagrange MVT for derivatives, integrals also have a mean value across an interval.

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