Riemanns Sums

4030 - Riemanns SumsAP Calculus

1A). AccumulationRate * Time = DistanceTRabArea under the curve represents the Accumulated DistanceBig Umbrella: Two Models:50 mphD=r*t50(3)=150accumulationArea under the curveDistance traveled or money2

abRTAccumulationWe dont drive at a constant rate due to lights, traffic, and etc. But the distance must be the same3Method of Exhaustion:Archimedes approximated by trapping the area of a circle between inscribed and circumscribed polygons

up to a 96-gon4Rectangular Approximations

5Rectangular Approximations - Development

A1A2A3A4b=xh=f (x)

6Under Approximations

Under Approximation: Inscribed Rectangles

7Over Approximations

Over Approximation: Circumscribed Rectangles

Under Approximation Total Area Over Approximation

8MethodA). 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. 9Ex 1:Find the Under and Over Approximations of the area under the graph of from 0 to 3 with x = 1.

A). Sketch the graph and the partitions on x.B). Sketch the height at each partition and find its height.C). Aunder =C). Aover =underover0 1 2 30 1 2 310Ex 2:Find the Under and Over Approximations of the area under the graph of from to with partitions at the friendly numbers

Aover =Aunder =

11EventTime(s)Velocity ft/sLaunch 0 0Begin Roll maneuver10 185End Roll maneuver15 319Throttle to 89%20 447Throttle to 67%32 742Throttle to 104%591325Maximum dynamic pressure621445Endeavor, Lift-off May 7,1992Find the Over and Under Approximations for the altitude of Endeavor at 62 sec.12B). Riemanns Sums

Xa

Xb

f (xa )f (x1)f (x2)f (xb )Riemann showed that any height in the subinterval could be used to approximate the accumulation.Definition : Riemanns Sum

is the length of the subinterval.is any point in the subinterval

13Riemanns LeftWe will use three of Riemanns sums.LEFT Riemann: using the LEFT partition always

1 2 3 4 514Riemanns RightWe will use three of Riemanns sums.RIGHT Riemann: using the RIGHT partition always

1 2 3 4 515Riemanns MidpointWe will use three of Riemanns sums.MIDPOINT Riemann: using the MIDPOINT of the partition always

NOTE: x does not change.

1 2 3 4 516Ex 3: Regular PartitionsFind the Left Riemanns, Right Riemanns, and Midpoint Riemanns approximations for the accumulation.

x = .

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Ex 4: Convenient PartitionsFind the Left Riemanns, Right Riemanns, and Midpoint Riemanns approximations for the accumulation.

18Pollution ControlThe 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. 0 0:30 1 1:302 2:30 3 3:30 4 4:30 5 5:30 6 1814 1735 1686 1646 1637 1609 1604 1611 1621 1666 1745 1886 2052TppiFind the Over and Under approximations for the quantity with t = 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 Riemanns and determine if sanctions are required.19Last Update:01/03/07

Assignment : worksheet 20Volume:Text #24 p. 272

21Volume:

22Riemanns Sums 2We will use three of Riemanns sums.LEFT RiemannRIGHT Riemann MIDPOINT Riemann

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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 RiemannsGraphical J F M A M J J A S25