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Definition of Integration (Reimann sums, etc.)

  1. May 22, 2012 #1
    So I was just working through Courant's calculus and am a bit confused as to where a few variables are pulled out of.

    1. The problem statement, all variables and given/known data

    Integration of f(x) = x

    We can see that a trapezoid is formed, so the relevant equation:
    1/2(b-a)(b+a) is the value of this integral.

    To confirm that our limiting process leads analytically to the same result, we subdivide the interval from a to b into n equal parts by means of the points of division
    a + h, a + 2h, . . ., a + (n-1)h, where h = (b-a)/n.
    (I still understand at this point as this is simply diving into n pieces)

    Taking for εi the right-hand end point of each interval we find the integral as the limit as n -> ∞ of the sum

    Fn = (a+h)h + (a+2h)h + . . . + (a + nh)h
    (At this point I am not sure where the h outside the brackets has come from and what it represents, I thought h was the distance between segments?)
    = nah + (1+2+3+ . . . + n)h2
    = nah + (1/2)n(n+1)h2
    (And at this point I am basically completely lost, I know the arithmetic series formula applies, but do not understand how to get to this point.)
     
  2. jcsd
  3. May 22, 2012 #2
    Never mind, I foiled it out and managed to figure it all out.

    It didn't help that I was mistakenly looking at the term of a series formula for arithmetic series rather than the sum formula :-)
     
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