1. AP Calculus Unit 5 Day 8

2. Area Problems Learning Outcome:  Combine integration techniques and geometry knowledge to determine total area

3. Review Problem …… 1. Set up an integral that represents the shaded region. 2. Evaluate the integral using the Fundamental Theorem of Calculus THEN confirm your answer with “ fnInt”

4. Find the area of the shaded region Discuss with your partner what we need to do to find the area of the shaded region. Important to Remember: Integrals find area between the curve and the x-axis.

5. Remember ….. When evaluating integrals, “areas” beneath the x- axis are negative. When evaluating total area, all areas are positive. a)Evaluate the integral a)Find the total area of the graph 124 f(x)

6. Practice—Find the area of the shaded region. (MINI HANDOUT) We will do #17 together Use fnInt to find the integrals

7. YOU TRY #18-20!! You may use fnINT for the integration.

8. Check your answers…

9. Questions About Area Problems Learning Outcome:  Combine integration techniques and geometry knowledge to determine total area

10. Next Learning Outcomes…  Apply concept of area to motion problems  Recognize the relationship between displacement and the total distance traveled by an object

11. Necessary Background Info For a position function,. So, the antiderivative of velocity is position, And the change in position over the interval (a,b) is represented by:

12. Summary--PVA Position Velocity Acceleration Derivatives Integration

13. Terminology ….. Distance Traveled—amount of movement by an object Displacement—how far away from the starting point an object is at the end of a given time interval

14. Building on the Area Concept Let’s assume represents the velocity of an object that is moving on a horizontal line. NOTE of Emphasis: This is the velocity curve NOT the path of the object.

15. Building on This Concept For what values of x is the object moving to the right? Therefore, represents the change in position in the “right” direction.

16. Building on This Concept What happens to the motion of the object at x=3? At x=3, how far has the object traveled? What is the displacement of the object? stationary 4.5 units 4.5 units to the right of start

17. Building on This Concept For what values of x is the object moving to the left? Therefore, represents the change in position in the “left” direction.

18. Building on This Concept From x=3 to x=4, how far has the object traveled? So, the TOTAL DISTANCE traveled would be: Using integral notation this would be: OR 11/6 units left 4.5+ 11/6=19/3

19. Building on This Concept From x=0 to x=3, the object moved 4.5 to the right. From x=3 to x=4, the object moved 11/6 back to the left. So, the DISPLACEMENT of the object would be: Using integral notation this would be: OR 4.5+ (-11/6)=8/3

20. Updating the Terminology Distance Traveled—amount of movement by an object equals the AREA between the velocity curve and the x-axis Displacement—how far away from the starting point an object is at the end of a given time interval equals the integral value of the velocity curve

21. MINI Handout #2: The graph above shows the velocity of a particle that is moving along a horizontal line for 20 seconds. Checking for UNDERSTANDING, fill-in the Blanks: The “area” above the x-axis represents the distance traveled in the _______________ direction. The “area” below the x-axis represents the distance travled in the _______________ direction. The total distance traveled can be represented by ________________________.

22. a) What is the total distance traveled from 0 to 20 seconds?

23. Another graphical option:

24. b) What is the displacement of the particle from 0 to 20 seconds?