[Math] Comparisons of Riemann Sums

calculus

Rank the following from lowest to greatest regarding the function: $f(x) = x^3 + 2x^2 + 5x + 6$ on the interval [0,1]

I. Left Sum using 1000 subintervals, II. Midpoint Sum using 1000 subintervals, III. Right Sum using 1000 subintervals, IV. Simpson's Rule using 1000 subintervals, V. Trapezoidal Rule using 1000 subintervals My logic goes as follows. I know the Left Sum is less than the Right Sum. But I also know that the Right Sum is greater than all the other sums and then visually, the Midpoint sum seems to be smaller than the Trapezoidal Sum. Would this logic be correct, or have I made a mistake somewhere?

Best Answer

  • If $f$ is increasing, we have $$f(x_n)<f\left( \tfrac{1}{2}(x_n+x_{n+1}) \right)<f(x_{n+1}),$$ so we have left sum < midpoint sum < right sum. This is reversed if $f$ is decreasing.
  • If $f$ is convex, $$ f\left( \tfrac{1}{2} (x_n + x_{n+1}) \right) \leq \frac{1}{2}(f(x_n)+f(x_{n+1})), $$ so the trapezium sum is larger than the midpoint sum. Moreover, the trapezium rule is an overestimate, the midpoint rule an underestimate (rotate the line to be tangent to the curve to see this for the midpoint rule: the trapezium formed has the same area as the original rectangle and lies wholly below the curve). This is reversed if $f$ is concave.
  • Simpson's Rule is the average of the trapezium and midpoint rules, so it must lie between them if they differ.

So all you have to do is show that $f$ is increasing and convex to determine the ordering.

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