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Math Help - Riemann Sums

  1. #1
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    Riemann Sums

    a) Approximate the distance traveled in the first three seconds by a particle with the velocity v(t) = (t^2 + 3t)/(t + 1). Use n = 4,10,20,50,100.

    b) Calculate the exact value using a definite integral.

    This problem doesn't seem that hard. You would just take the anti-derivative of the equation then where t is, divide t by n?
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  2. #2
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    Quote Originally Posted by BarlowBarlow1 View Post
    a) Approximate the distance traveled in the first three seconds by a particle with the velocity v(t) = (t^2 + 3t)/(t + 1). Use n = 4,10,20,50,100.
    A Reimann sum over an interval [a,b] corresponding to a=x0<x1< ... <xn=b is:

    S = sum_{i=1 to n} v(y_i) (x_i - x_{i-1})

    where y_i is some value in the interval [x_{i-1}, x_i]. If y_i=x_{i-1} we
    have a left Reimann sum and if y_i=x_i we have a right Reimann sum.

    Now here I think you are expected to calculate one of these where the
    x_i's are regularly spaced, with interval lengths h=(b-a)/n, then the left
    and right Riemann sums are:

    S_l = h * sum_{i=0 to n-1} v(a+i*h)

    S_r = h * sum_{i=1 to n} v(a+i*h)

    Now I presume that you have some computational tool to do this on.

    b) Calculate the exact value using a definite integral.

    This problem doesn't seem that hard. You would just take the anti-derivative of the equation then where t is, divide t by n?
    To do the integral exactly observe that:

    v(t) = (t^2 + 3t)/(t + 1) = t + 2 - 2/(t+1)

    which is elementary.

    RonL
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