6.1 Estimating with Finite Sums AP Calculus AB 6.1 Estimating with Finite Sums

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.

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:

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

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). In this example there are four subintervals. As the number of subintervals increases, so does the accuracy. Approximate area:

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

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

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-83/TI-84 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.

If you have the AREA program installed: Set up the screen as follows: WINDOW

Press Y = Enter: Press PRGM Select 1:AREA and press ENTER Press ENTER again to run the program. Make the Number of Intervals: 4 Make the x Value: 0 Make the Upper x Value: 4

p*

Volume of a Nose Cone. The nose “cone” of a rocket is a paraboloid obtained by revolving 0≤ x ≤5 about the x-axis. Estimate the volume V of the nose cone by partitioning [0, 5] into five subintervals of equal length, slicing the cone with planes perpendicular to the x-axis at the sub intervals’ left endpoints, constructing cylinders of height 1 based on cross sections at these points, and finding the volumes of these cylinders. Exercises 24 and 25, page 272.

Press Y = Enter: Press PRGM Select 1:AREA and press ENTER Press ENTER again to run the program. Make the Number of Intervals: 5 Make the x Value: 0 Make the Upper x Value: 5

Actual Value π