1. 8.1 A – Integral as Net Change Goal: Use integrals to determine an objects direction, displacement and position.

2. Position Position is measured in two ways, displacement and distance

3. Displacement Distance The difference between the starting and ending positions of an object (easy) The total amount of “ground covered” by an object or the total length of its path (more difficult)

4. Distance = Displacement = If you were to drive 10 miles east and then 4 miles west 10 miles east 4 miles west 14 miles 6 miles Example

5. velocity graph Displacement: Distance Traveled: Example looking at a graph

6. Simply integrate velocity. The negative areas below the x-axis subtract from the total displacement. Displacement

7. To find distance traveled we have to use absolute value. Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.) Or you can use your calculator to integrate the absolute value of the velocity function. Distance

8. Example Let v(t) = 2t 3 – 14t 2 + 20t, Determine when the particle is moving to the right, left, and stopped. Find the particle’s distance and displacement after 3 seconds. v(t) = 2t 3 – 14t 2 + 20t v(t) = 2t(t 2 – 7t + 10) v(t) = 2t(t – 2)(t – 5) 0 2 5 0 ++0 ------- 0 +++++ Right (0,2) U (5,  ) Left (2,5) Stopped t = 0, 2, 5

9. Continued... Find displacement….

10. Continued... Find distance….

11. You try! A particle moves along the x-axis so that its acceleration at any time t is given by a(t) = 6t - 18. At time t = 0 the velocity of the particle is v(0) = 24, and at time t = 1 its position is x(1) = 20.

12. Questions: (a) Write an expression for the velocity v(t) of the particle at any time t. (b) For what values of t is the particle at rest? (c) Write an expression for the position x(t) of the particle at any time t. (d) Find the total distance traveled by the particle from t = 1 to t = 3.

13. Answer part a

14. Answer part b The particle is at rest when v(t) = 0. 3t 2 - 18t + 24 = 0 t 2 - 6t + 8 = 0 (t - 4)(t - 2) = 0 t = 4, 2

15. Answer part c

16. Answer part d

17. Homework