Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sum of ordinates mean value of functions

  1. Nov 18, 2014 #1
    I am having trouble deciphering the opening gambit of an explanation of mean values of functions. It begins as follows:

    "Consider the part of the curve y = f(x) for values of x in the range a ≤ x ≤ b."

    A graph is shown with a curve cutting the x axis at c with a shaded positive area bounded by the curve and the line x=a to the left of c and a shaded negative area bounded by the curve and the line x = b to the right of c.

    "The mean value of y in this range is the average value of y for that part of the curve.
    The sum of the ordinates (ie values of y) between x= a and x = c occupies the shaded area above the x axis and is positive.
    This area is ∫acy dx
    Hence the sum of the ordinates between x = a and x= c is ∫acy dx"

    I understand that an ordinate is the value of y. But are the ordinates taken at integer values of x or continuous values of x. I don't see how the sum of ordinates is equal to the value of the area under the curve. I must have misunderstood the definition of the sum of the ordinates.
    I can see how the sum of the continuous ordinates multiplied by change in x as change in x goes to 0 might equal the area under the curve.

    Sorry if my terminology and description of the graph leave a lot to be desired.
     
  2. jcsd
  3. Nov 18, 2014 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    It's indeed meaningless to speak of the sum of all the y coordinates of the graph of a function that is defined on interval of real numbers. unless you define "sum" to be something besides an ordinary arithmetic sum.

    To argue the relation between an integral and a mean value in a better way, consider that "mean mass per unit length" is defined by a relation such as (total length of interval )(mean mass per unit length) = total mass

    Think of a f(x) as being "mass density" Then you just need to understand why the integral of a mass density over an interval is the total mass of the interval.

    You book could have said "Think of [itex] f(x) [/itex] as being a density of something. Then [itex] \int_a^b f(x) dx [/itex] is the total something in the interval [itex] [a,b] [/itex] and [itex] \frac{ \int_a^b f(x) dx}{ [b-a]} [/itex] is the mean something per unit length = the mean density.
     
  4. Nov 20, 2014 #3
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Sum of ordinates mean value of functions
  1. Sum of Functions (Replies: 3)

Loading...