1. SECTION 5.1: ESTIMATING WITH FINITE SUMS Objectives: Students will be able to… Find distance traveled Estimate using Rectangular Approximation Method Estimate the Volume of a sphere

2. DISTANCE TRAVELED If you were traveling at a constant rate of 60 mph for 3 hours, how far did you travel? Let’s look at a graph, shall we???

3. RECTANGULAR APPROXIMATION METHOD (RAM)  Can find total area under the curve by dividing up the interval into subintervals (little rectangles) and adding up the area of each subinterval (rectangle )  If you have a function over interval [a,b]: Width of rectangle=, where n is the number of subintervals. Height of rectangle = the value of the function at either the right endpoint, left endpoint, or midpoint of the interval  As n →∞, the more accurate your approximation

4. RECTANGULAR APPROXIMATION METHOD  LRAM : left endpoint RAM  RRAM : right endpoint RAM  MRAM : midpoint RAM

5. The rate of sales (in games per week) of a new video game is shown in the table below. Assuming that the rate of sales increased throughout the 20-week period, estimate the total number of games sold during this period. Time weeks 05101520 Rate of sales games/week 058589218752350

6. A car is moving with increasing velocity and we measure the car’s velocity every two seconds. How far has the car traveled? Time (sec) 0246810 Velocity (ft/sec) 203038444850

7. EXAMPLE: The velocity in m/s of an object moving along a straight line is given by the function v=t 2, where 0 < t < 8. Approximate the displacement of the object by dividing the interval [0, 8] into n subintervals of equal length. Use n= 8, and MRAM.

8. LOOK AT THE GRAPH OF THE FUNCTION Y = -X 2 +4 ON [0, 2].  Divide into 8 subintervals  Is the function inc or dec over interval?  Will RRAM be overestimate or underestimate?  LRAM? Find the area under the curve using RRAM.

9. GENERAL GUIDELINES  INCREASING FUNCTION: RRAM overestimates, LRAM underestimates  DECREASING FUNCTION : RRAM underestimates, LRAM overestimates.