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Integral as Accumulation

Integral as Accumulation. What information is given to us by the area underneath a curve?. You are given a velocity function where velocity is constant . Find the distance traveled at any time, . Velocity and Distance. Since we know: D istance = Velocity Time. 3.

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Integral as Accumulation

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  1. Integral as Accumulation What information is given to us by the area underneath a curve?

  2. You are given a velocity function where velocity is constant . Find the distance traveled at any time, . Velocity and Distance Since we know: Distance = Velocity Time • 3 So we can clearly see that the distance traveled is equal to the area under the graph of this constant velocity function.

  3. Velocity and Distance What if velocity is not constant? Will the area underneath the graph still represent the distance? To answer this, again think about the area as a Riemann Sum.

  4. To think about this, we can approximate the area over any interval by using a Riemann Sum and initially allowing The velocity of a function is given as . What is the distance traveled from to ? (We will use a midpoint riemann sum where the height will represent an approximation of the average velocity over this interval). + The area of each rectangle would represent the distance traveled over that time interval if we traveled at a constant velocity represented by the height. To make this more exact, we can increase the number of rectangles and divide the time into smaller and smaller pieces.

  5. Distance as Integral of Velocity When given the graph of velocity and asked to find the distance, we can divide up the area into rectangles where the area of each will represent the distance traveled. If we allow , we will get closer and closer to the actual distance. We already know the area under a graph can be found by integrating the function and therefore….

  6. Average Velocity and Distance Use the Velocity function, to find the average velocity from . -We know the average velocity from is equal to 7. -Therefore if we travel for 3 hours. -We know the distance traveled will be 21 (the same as the area under the graph).

  7. Distance vs. Position I used distance in order to simplify things. We can really only use distance in the previous slides when the area is always above the Consider the distance traveled in this function. • If… Then the distance traveled (assuming ) would be 0. Since velocity is not always 0, this makes no sense.

  8. The correct terminology is… So the change in Position (assuming is 0. The particle moved to the right units and then moved to the left B units. Since , the particle is where it started and the change in position is 0. The distance traveled *Position is also sometimes referred to as displacement.

  9. Integration of any rate of change can be used to find the Net Change. • If you are given a graph or function that represents the rate of flow of water (liters/hour), the integral will give you the total amount or net change of water. • If you are given a rate of growth/decline of a population (people/year), then the integral will give you the net change in population.

  10. The area of the rectangle….. Velocity and Position • Population Growth Water Flow Net Change in Position (Displacement) • Water • Flow • Population • Growth • Net Change in Population Velocity Net water flow • Time Time • Time

  11. Water Flow Water flows out of a cylindrical pipe at a rate given by liters/second, After 20 seconds, how much water has left the pipe?

  12. Homework Net Change/Accumulation Worksheet (1, 3, 12-16, 19, 21, 27)

  13. Initial Position When finding the current position, you must know where something started. Imagine a particle moved to the left 5 units. It’s final position will depend upon its initial position.

  14. Give that g, find the following:

  15. If a graph of acceleration is given, what would the integral represent? Acceleration and Velocity The area under the graph would represent the Net Change in velocity. If the initial velocity, , Calculate the following velocities…

  16. Think about the Rectangle

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