The Definite Integral as an Accumulator

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Presentation transcript:

The Definite Integral as an Accumulator Bob Arrigo Scarsdale High School Scarsdale, NY r1arrigo@gmail.com www.BCCalculus.com

Traditional applications of the Definite Integral prior to the Calculus reform movement Area, volume, total distance traveled.. (AB) Arc length, work.. (BC) Mass, fluid pressure.. (Some college Calculus courses)

Calculus Reform in the early 90’s brought in “broader”, more robust applications of the definite integral…… most prominently, use of the definite integral to calculate “net change”, or “accumulated change.”

Types of Integrals Definite Integrals…limits of Riemann sums …”summing up infinitely many infinitesimally small products” Indefinite Integrals….a family of functions Integral functions….functions defined by an integral

The definite integral provides net change in a quantity over time. The definite integral of a rate function yields accumulated change of the associated function over some interval.

Motivate with a water flow problem: The rate at which water flows into a tank, in gallons per hour, is given by a differentiable function R of time t. Values of R are given at various times t during a 24 hour period. Approximate the number of gallons of water that flowed into the tank over the 24 hour period. t R(t) 13 6 15 12 18 14 24 10

This is an approximation for the total flow in gallons of water from the pipe in the 24-hour period.

Summing up lots of distances, each of which equals the product (rate)(time)

Method I to get the total distance traveled: Break up the interval [0,6] into smaller and smaller subintervals. To get the actual distance traveled, use more, smaller subintervals.

t v(t) 2 7.2 4 12.8 6 16.8 Method II

Since method I and method II, both yield total distance, We get: Answer method I = Answer method II

Since method I and method II both yield total distance, We get: Answer method I = Answer method II

End Amt = Start Amt + NET CHANGE

Since is positive for and is negative for, the maximum value for occurs at time.

Since is positive for and is negative for, the maximum value for occurs at time.