5.5 Area as Limits Greenfield Village, Michigan Greg Kelly, Hanford High School, Richland, Washington Modified: Mike Efram Healdsburg High School 2004 Photo by Vickie Kelly, 2002

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 value for the area is when we use n subintervals, as n→∞. With 8 subintervals: Approximate area: The exact value for the area is when we use n subintervals, as n→∞. width of subinterval

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

Area under the curve As the interval ∆x gets smaller, we have more rectangles, and thus have a better approximation of the area. If we take the limit as ∆x→0 (or n→∞), we will find the actual area.

With n subintervals: Exact area:

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.