Math (Urgent)

Use the Midpoint Rule to approximate the integral (–10x+7x^2) from 10 to 17 with n=3

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  1. Hint:
    Integration of f(x) by the mid-point rule for the interval [a,b].
    1. subdivide [a,b] into n sub-intervals, each of length h=(b-a)/n.
    2. Evaluate the value qi of the function f(x) at the mid-point of each sub-interval i, where i=1,n. That is,
    qi = f(a+(i-1/2)h).
    3. Calculate the sum:
    h∑qi for i=1 to n.

    Example:
    If f(x)=x²
    We would like to approximate
    I=∫f(x)dx for x= [20,28] with n=4.
    h=(28-20)/4=2
    q1=f(20+h/2)=f(21)=441
    q2=f(20+3h/2)=f(23)=529
    q3=f(20+5h/2)=f(25)=625
    q4=f(20+7h/2)=f(27)=729
    I=(441+529+625+729)*2=4648 (approx.)

    Check by exact calculations:
    ∫x²dx
    = (28³-20³)/3
    = 13952/3
    = 4650.7 (approx.)
    So the answer by the mid-point rule is quite close to the analytic value.

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