Compare left Riemann sums, right Riemann sums, and midpoint sums for the same function

Primary Tool

Spreadsheet Workspace

Spreadsheet is the right primary tool because the point is comparative structure: the same intervals, different sample points, and different totals. That is much easier to read in columns than in prose alone.

Problem

Compare the left, right, and midpoint Riemann sums for $f (x) = x^{2}$ on $[0, 1]$ using $4$ equal subintervals.

This is a good comparison problem because the exact integral is known, so the approximations can be judged against a real answer rather than against each other only.

What the sample point changes

All three sums use the same partition width $Δ x = 1/4$ . What changes is only the sample point chosen inside each subinterval.

For an increasing function like $x^{2}$ on $[0, 1]$ , the left sum tends to underestimate and the right sum tends to overestimate. The midpoint sum often lands closer because it samples more centrally in each interval.

In this specific table, the totals come out to $L_{4} = 0.21875$ , $R_{4} = 0.46875$ , and $M_{4} = 0.328125$ , while the exact integral is $1/3 \approx 0.3333$ . That makes the midpoint improvement visible instead of merely theoretical.

Step-by-step walkthrough

The spreadsheet puts the three approximation methods side by side so the only moving part is the sample-point choice.

1Partition $[0, 1]$ into $4$ equal intervals, so $Δ x = 0.25$ .
2Use the left endpoints to compute the left sum $L_{4}$ .
3Use the right endpoints to compute the right sum $R_{4}$ .
4Use the interval midpoints to compute the midpoint sum $M_{4}$ .
5Add the rectangle areas in each column to compare the three totals.
6Compare those totals with the exact integral $\int_{0}^{1} x^{2} d x = 1/3$ .

Common Pitfall

A Riemann sum is not one fixed formula. The partition width can stay the same while the approximation changes noticeably depending on whether you choose left endpoints, right endpoints, or midpoints.

Try a Variation

Repeat the comparison for $f (x) = x$ on $[0, 1]$ . Which sums overestimate or underestimate now, and how does the graph explain the change?

Keep moving through the cluster

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