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

Velocity and Distance So we can clearly see that the distance traveled is equal to the area under the graph of this constant velocity function.

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.

To make this more exact, we can increase the number of rectangles and divide the time into smaller and smaller pieces. The area of each rectangle would represent the distance traveled over that time interval (We will use a midpoint riemann sum where the height will represent an approximation of the average velocity over this interval). if we traveled at a constant velocity represented by the height.

Distance as Integral of Velocity We already know the area under a graph can be found by integrating the function and therefore….

Average Velocity and Distance

Distance vs. Position Consider the distance traveled in this function. Since velocity is not always 0, this makes no sense. If…

The correct terminology is… *Position is also sometimes referred to as displacement.

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.

The area of the rectangle….. Velocity and Position Water Flow Population Growth

Water Flow

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

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.

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

Think about the Rectangle