Acme adding machine

Properties of Integrals

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

Also recall:

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.

and ...

Use The Fundamental Theorem of Calculus to evaluate:

Properties of Integrals ...

Constant Multiple Rule

Sum and Difference Rule

Additive Interval Rule

Inequality Rule

Let's see some examples ...

If ƒ(x) ≤ g(x) for all x

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

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

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-cetO Integrals +

–=

Build your own equations from the bucket:

= +

= –

-3

4 = -3 – 4

Inequality Rule Consider each of these integrals and the related graphsIf ƒ(x) ≤ g(x) for all x

The Definite Integral of a Rate Give Total ChangeLuke! Remember the FTC!

obi-wan kenobi

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.