4030 - Riemann’s Sums

1. 4030 - Riemann’s Sums AP Calculus

2. A). Accumulation R Rate * Time = Distance 50 mph D=r*t 50(3)=150 T a b Area under the curve represents the Accumulated Distance Big Umbrella: Two Models: accumulation Area under the curve Distance traveled or money

3. Accumulation R We don’t drive at a constant rate due to lights, traffic, and etc. But the distance must be the same T a b

4. Method of Exhaustion: Archimedes approximated π by trapping the area of a circle between inscribed and circumscribed polygons up to a 96-gon

5. Rectangular Approximations

6. Rectangular Approximations - Development h=f (x) A1 A2 A3 A4 b=x

7. Under Approximations Under Approximation: Inscribed Rectangles

8. Over Approximations Over Approximation: Circumscribed Rectangles Under Approximation  Total Area  Over Approximation

9. Method A). Sketch the graph and the partitions on x. B). Sketch the height at each partition and find its height. C) Choose the value required for the approximation. xc = Circumscribed rectangle for OVER Approximation xi = Inscribed rectangle for UNDER Approximation and multiply by the base ( x ) for the Area of each rectangle. D) Add the areas.

10. Find the Under and Over Approximations of the area under the graph of from 0 to 3 with x = 1. Ex 1: over under A). Sketch the graph and the partitions on x. 0 1 2 3 0 1 2 3 B). Sketch the height at each partition and find its height. C). Aunder = C).Aover =

11. Find the Under and Over Approximations of the area under the graph of from to with partitions at the friendly numbers Ex 2: =1 Aunder= Aover=

12. Endeavor,Lift-off May 7,1992 Event Time(s) Velocity ft/s Launch 0 0 Begin Roll maneuver 10 185 End Roll maneuver 15 319 Throttle to 89% 20 447 Throttle to 67% 32 742 Throttle to 104% 59 1325 Maximum dynamic pressure 62 1445 Find the Over and Under Approximations for the altitude of Endeavor at 62 sec.

13. f (xa ) f (x1) f (x2) f (xb ) B). Riemann’s Sums Riemann showed that any height in the subinterval could be used to approximate the accumulation. Xa Xb Definition : Riemann’s Sum is the length of the subinterval. is any point in the subinterval

14. Riemann’s Left We will use three of Riemann’s sums. LEFT Riemann: using the LEFT partition always 1 2 3 4 5

15. Riemann’s Right We will use three of Riemann’s sums. RIGHT Riemann: using the RIGHT partition always 1 2 3 4 5

16. Riemann’s Midpoint We will use three of Riemann’s sums. MIDPOINT Riemann: using the MIDPOINT of the partition always NOTE: xdoes not change. 1 2 3 4 5

17. Ex 3: Regular Partitions x = ½. Find the Left Riemann’s, Right Riemann’s, and Midpoint Riemann’s approximations for the accumulation.

18. 1 Ex 4: Convenient Partitions Find the Left Riemann’s, Right Riemann’s, and Midpoint Riemann’s approximations for the accumulation. .5

19. The TABLE shows the rate of emissions of pollutants from a plant from 12 midnight to 6 am. The EPA regulates the quantity of pollutants and assesses a fine if the quantity is over 10,000. Pollution Control T ppi 0 0:30 1 1:30 2 2:30 3 3:30 4 4:30 5 5:30 6 1814 1735 1686 1646 1637 1609 1604 1611 1621 1666 1745 1886 2052 Find the Over and Under approximations for the quantity with t =½ hr. The plants officials use the under approximation to argue that their emissions are with in the standards. The environmental advocates use the over approximation to argue for sanctions. the EPA, required to make a decision, wants a better estimate. Use the Left and Right Riemann’s and determine if sanctions are required.

20. Last Update: • 01/03/07 • Assignment : worksheet

21. Volume: Text #24 p. 272

22. Volume:

23. Riemann’s Sums 2 We will use three of Riemann’s sums. LEFT Riemann RIGHT Riemann MIDPOINT Riemann

25. Graphical The rate of exchange of the dollar versus the Euro over time graph is given. Find the net value of the investment during the 8 months using four subintervals and a) Left Riemanns,(b) Right Riemanns,and (c) Midpoint Riemanns J F M A M J J A S