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Calc I - Finding average velocity using integration

  1. Oct 29, 2005 #1
    Can I use integration to show why car 2 traveled farther than car 1 over a given time interval?

    Here is some of the problem,

    I have the following graph of two different velocity functions
    for two cars.

    TI-89 Graph

    The viewing window of this graph is x from 0 to 30
    and y from 0 to 100.

    After integrating the velocity function I found the position function.
    Estimating the distances traveled by both cars from time t = 0
    to time = 30 I find,

    Car 1 whose velocity function is the thin line on the graph
    traveled approximately 964.11 feet.

    Car 2 whose velocity function is the thick line on the graph
    traveled approximately 1977.9 feet.

    By looking at the graph it is obvious that the thick line has
    a larger region under the curve (from the curve to the x-axis) than
    the thin line.

    Is it possible to integrate from 0 to 30 on each velocity function and show that the larger regions under the curve correspond to a higher average velocity over time interval 0 to 30 which is why car 2 traveled farther? Am I on the right track about dealing with the velocities of the cars? Should I also consider their acceleration as being a reason for why car 2 traveled farther?

    I need explain why car 2 traveled farther. Is it possible to use integration to find average velocity?

    In the original problem velocity is given every 5 seconds from 0 to 30 for each car. I could find the secant line over each 5 second interval and average those secant lines to find the approximate average velocity from time 0 to 30, correct? If possible I want to try to use integration to solve this.

    Thanks
     
    Last edited: Oct 29, 2005
  2. jcsd
  3. Oct 29, 2005 #2
    You have two velocity graphs and you want to find the average velocity. Your instincts are correct to integrate, but it's not just integrating from 1 to 30. Calculating the average value of a function, f(x), from a to b can be done as follows:

    [tex]A(x)=\frac{1}{b-a}\int_{a}}^{b}f(x)dx[/tex]
     
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