7.1 Integral as Net Change

1. 7.1 Integral as Net Change Greg Kelly, Hanford High School, Richland, Washington

2. Fiona is very proudly showing her King of the Hill car to the class. Diabolically, however, Andy has secretly attached a small motor to the car to make it zig-zag by remote control. While Fiona futilely chases the car back and forth, the car goes first north at 10 ft/sec, stops, then south at 10 ft/sec and continuing this torturous pattern over a period of 30 seconds Manisha is barely able to stop laughing long enough to make a graph of the car’s motion…

3. The velocity graph is shown below. What is the total distance traveled by the car? 200 feet 10 ft/sec north 10 ft/sec north 50 ft 100ft v = 0 v = 0 50 ft 10 ft/sec south

4. The velocity graph is shown below. What is the displacement of the car? 100 feet north 10 ft/sec north 10 ft/sec north 50 ft 100ft v = 0 v = 0 –50 ft 10 ft/sec south

5. Change in position is measured in both displacement and distance The difference between the starting and ending positions of an object The total length of the object’s path If you were to drive 10 miles east and then 4 miles west 10 miles east 4 miles west Displacement = 6 miles Distance = 14 miles

6. Displacement: velocity graph Distance Traveled: |velocity| graph

7. position graph Displacement: A displacement of –1 is not the same as a displacement of 1. The difference here is…? velocity graph direction Distance Traveled:

8. In terms of driving a car on a round trip, would be like asking how far it was from where your car started to where it ended the trip. would be like asking how much mileage you put on your car during the round trip. Your odometer wouldn’t be accounting for which direction you were going.

9. The rate of potato consumption for a particular country was: where t is the number of years since 1970 and C is in millions of bushels per year. This same technique of integrating rates of change to find quantities is used in many different real-life problems. National Potato Consumption

10. million bushels National Potato Consumption How would we find the total consumption? How about specifically from the beginning of 1972 to the end of 1973: end of ‘73 (beginning of ‘74) beginning of ‘72

11. million bushels National Potato Consumption By the way, what is a great way to make sure that we are on the right track here? Units. Why? We go from millions of bushels/year to millions of bushels. How about specifically from the beginning of 1972 to the end of 1973:

12. National Potato Consumption By the way, what is a great way to make sure that we are on the right track here? Units. Why? We go from millions of bushels/year to millions of bushels. How about specifically from the beginning of 1972 to the end of 1973: There will be other problems like this that involve calculating quantity of product from some rate of production or consumption.

13. And finally, remember… In terms of driving a car on a round trip, would be like asking how far it was from where your car started to where it ended the trip. would be like asking how much mileage you put on your car during the round trip. Your odometer wouldn’t be accounting for which direction you were going.