The document discusses properties of integrals including: 1) The Fundamental Theorem of Calculus which states that integrating the derivative of a function over an interval gives the total change in the function over that interval. 2) Rules for integrals including the additive interval rule, constant multiple rule, sum and difference rule, and inequality rule. 3) Examples are provided to illustrate applying these rules to evaluate integrals.

1. Properties of Integrals
Acme adding machine

2. What does The Fundamental Theorem of Calculus say?
Let the function ƒ be continuous on [a, b] with derivative ƒ'. Then ...
Total change in ƒ
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:
and ...

3. Use The Fundamental Theorem of Calculus to evaluate:

4. Properties of Integrals ...
Additive Interval Rule
Constant Multiple Rule
Sum and Difference Rule Inequality Rule
If ƒ(x) ≤ g(x) for all x
Let's see some examples ...

5. Constant Multiple Rule Consider each of these integrals and the
related graphs

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

7. Additive Interval Rule This rule is illustrated in the graph below.
It allows us to solve problems like this
one:
Given: and
Find:
Ye Olde Buh-cet Build your own equations from the bucket:
-3 O Integrals + = +
–
=
= –
–
= -3 4
4

8. Consider each of these integrals and the
Inequality Rule
related graphs
If ƒ(x) ≤ g(x) for all x

9. The Definite Integral of a Rate Give Total Change
Luke! Remember the FTC!
A company estimates that its rate of sales of an
item is given by:
Where t is measured in years. Find the total sales
during the years 2 ≤ t ≤ 4.
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