1. 5.1 Estimating with Finite Sums Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 Greenfield Village, Michigan

2. time velocity After 4 seconds, the object has gone 12 feet. 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.

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

7. Circumscribed rectangles (or upper sum) are all above the curve: Inscribed rectangles (or lower sum) are all below 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.

9. Use a midpoint Riemann sum with three subintervals of equal length indicated by the data in the table to approximate the area under the curve from t = 10 to t = 70. Using correct units, explain the meaning of this area in terms of the rocket’s flight.

10. The graph of the velocity v(t), in ft/sec, of a car traveling on a straight road, for 0 ≤ t ≤ 50, is shown above. Approximate the area under this curve with a Riemann sum, using the midpoints of five subintervals of equal length. Explain the meaning of this area.

11. The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. Use a midpoint Riemann sum with 4 subintervals of equal length to approximate the area under the curve from 0 to 24. Explain the meaning of this area in terms of water flow.

12. Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate the area under the curve. Using correct units, explain the meaning of this area in terms of the radius.