Section 6.4
Let f(t) be a continuous function
on the closed interval [a, b]. If I
divide the interval into n equal subintervals, each piece will be length 
For
each subinterval, pick a point. (If I were using the right-hand method, I would
pick the right endpoint of the interval.)
So out of interval I, pick the point xi. At that point, I will go up to the function
and build a rectangle whose height is f(xi). The area of that rectangle is
. We can add all the
areas of all of the rectangles: 
. So if we let the
number of rectangles increase without bound we have
This is called the definite integral of f(x) with limits of integration a and b.
Example:
This is the same as asking the area between f(t) = t + 3 and the t-axis on the interval [1, 4].
This is the area of the trapezoid. We can think of this as a rectangle (base 3, height 4) topped by a triangle (base 3, height 3). So the area is 12 + 9/2 = 33/2
Example: Evaluate 
Graph the function f(t) = 3t + 2 on the interval [0, 5]