Today's class If we start off with a function f(x) we know how to find the derivative (slope) of the function.
The converse is to be given a function and ask the question: ``what was differentiated to get this function as the derivative''?
Example: what function would have as it's derivative 
?
Answer: 
where c is any constant.
We call the anti-derivative of .
Motivation for the integral.
Consider the velocity graph.
For a small change in time the area in the rectangle equals the distance you have traveled in that time period.
The sum of the areas of the rectangles approximates how far you have traveled during the entire time period.
As the time intervals get shorter the area under the velocity curve indicates how far you have traveled.
We know the rate of change of distance is velocity.
Therefore the anti-derivative of velocity is distance.
The area under the velocity graph is distance. Therefore to find the area under the velocity function (distance traveled) you need to find the ``anti-derivative'' of the velocity function.
To find the area under the graph of f(x) you need to find the anti-derivative of f(x).
If f(x) is the function we are interested in, and F(x) is it's anti-derivative then the area under the graph of f(x) between the points a and b is given by
We write the area under the curve as
Therefore
where F is the anti-derivative of f.
Examples:
What's the area under the curve 
between
the points a = 0 and b = 1?
Find
We need to find the anti-derivative of
?
Answer: 
. This is the
anti-derivative, F(x).
Therefore the area under the curve is F(1) - F(0) which equals
.
This one's easy because the function defines a straight line and the area is a triangle.
Example: what's the area under the curve of 
between the
points 0 and 2?
The anti-derivative is 
The area is by definition
.
Interpreting the area under the curve.
First note what units the area is measured in. The units of x times the units of y.
For the velocity graph, x is measured in seconds, and velocity (y) in
meters per second. Therefore the area under the velocity curve is
measured in seconds 
meters/seconds = meters, or
in other words distance.
The area under a marginal cost curve plotted against volume is in dollars. It tells you the total cost to produce that many units.
Examples from the old exams.