The Fundamental Theorem of Calculus states that integrating the derivative of a function over an interval gives the total change in the function over that interval, or the integral of a derivative equals the total change in its parent function. The document then provides examples of using the Fundamental Theorem of Calculus to evaluate integrals and discusses properties like the constant multiple rule, additive interval rule, sum and difference rule, and inequality rule.

1. What does The Fundamental Theorem of Calculus say?
Let the function ƒ be continuous on [a, b] with derivative ƒ'. Then ...
In words: Integrating a rate of change function ƒ' over an interval [a, b]
gives the total change in ƒ, ƒ(b) - ƒ(a), over the same interval.
In other words, the integral of a derivative is the same thing as
the total change in it's parent function over the same interval.
Also recall:

2. Use The Fundamental Theorem of Calculus to evaluate:

3. Properties of Integrals ...
Constant Multiple Rule Additive Interval Rule
Sum and Difference Rule Inequality Rule
Let's see some examples ...

4. Constant Multiple Rule
related graphs

5. Sum and Difference Rule Combining this rule with the
constant multiple rules allows us
to integrate expressions like this:

6. Additive Interval Rule
It allows us to solve problems like this
Ye Olde Buh-
Build your own equations from the bucket:
cet
O Integrals

7. Inequality Rule
related graphs

8. A company estimates that its rate of sales of an
item is given by:
is measured in years. Find the total sales
during the years 2 ≤