1. Lesson 5-1: Estimating with Finite Sums
AP Calculus
Mrs. Mongold

2. Consider an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance:
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
time
velocity
After 4 seconds, the object has gone 12 feet.

3. If the velocity is not constant,
we might guess that the
distance traveled is still equal
to the area under the curve.
(The units work out.)
Example:
We could estimate the area under the curve by drawing rectangles touching at their left corners.
This is called the Left-hand Rectangular Approximation Method (LRAM).
Approximate area:

4. We could also use a Right-hand Rectangular Approximation Method (RRAM).
Approximate area:

5. Another approach would be to use rectangles that touch at the midpoint
Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM).
Approximate area:
In this example there are four subintervals.
As the number of subintervals increases, so does the accuracy.

6. The exact answer for this problem is .
With 8 subintervals:
Approximate area:
The exact answer for this
problem is
width of subinterval

7. Inscribed rectangles are all below the curve:
Circumscribed rectangles are all above the curve:

8. We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation.
The TI-89 calculator can do these rectangular approximation problems. This is of limited usefulness, since we will learn better methods of finding the area under a curve, but you could use the calculator to check your work.