Calculus explainer
Rectangles to Area
Cut the interval from a to b into n equal strips, build a rectangle (or trapezoid) on each, and add up their areas. As n grows, the sum closes in on the exact area, the definite integral ∫ab f(x) dx.
- the graph of f
- area counted as positive (above the x-axis)
- counted as negative (below the axis)
Choose
The numbers
- Width Δx = (b − a)/n
- Left sum
- Exact integral
- Error (sum − exact)
- Relative error
How fast the error shrinks
- Left
- Right
- Midpoint
- Trapezoid
Both scales are logarithmic: each grid step is 10 times bigger. A line that drops one step for each step right has error about 1/n; a line twice as steep has error about 1/n².
| n | Left | Right | Midpoint | Trapezoid |
|---|
What to notice
Each rectangle has width Δx and height f at one sample point: the left end, the right end or the middle of its strip. The trapezoid rule joins the two ends with a slanted top instead, and equals the average of the left and right sums.
Where f is increasing, left sums come out too small and right sums too large; where it is decreasing, the reverse. Below the x-axis, area counts as negative.
Why it works
The definite integral is the limit of these sums as n → ∞. The Fundamental Theorem of Calculus gives its exact value from an antiderivative: ∫ab f(x) dx = F(b) − F(a).
Doubling n roughly halves the error of left and right sums, but cuts the midpoint and trapezoid errors to about a quarter. Watch the error graph.
Archimedes found the area of a parabolic segment more than 2,000 years ago by filling it with ever smaller pieces; Bernhard Riemann gave the modern definition in 1854.
Try it
- For x² on [0, 2], find the left, right and midpoint sums for n = 4 by hand, then check.
- Which is bigger for eˣ, the midpoint sum or the trapezoid sum? Why does the shape of the curve decide?
- Set x³ − 3x on [−2, 2]. What is the exact integral, and why?
- For 1/x on [1, 4], how large must n be before the left sum is within 0.01 of ln 4?
- From the table, divide the error at n = 10 by the error at n = 20 for each method. What ratios do you get?
Sums are added term by term from the formula for f; exact values come from the antiderivative. No outside code, images or trackers.