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Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.
Key values: x^2 · [0, 1] · Simpson's rule
Integrate sin(x) from 0 to pi. Exact answer: 2.
Key values: sin(x) · [0, pi] · Exact: 2
Approximate the Gaussian integral e^(-x^2) from -3 to 3.
Key values: e^(-x^2) · [-3, 3] · Gaussian
Enter a function f(x) with bounds and click Calculate to see the numerical integral, error estimate, and interactive Riemann sum visualization.
A Riemann sum approximates the area under a curve by dividing the region into rectangles. The interval is split into subintervals of width :
where is the sample point in the -th subinterval.
| Type | Sample point | Formula | Tendency |
|---|---|---|---|
| Left | Underestimates (if f increasing) | ||
| Right | Overestimates (if f increasing) | ||
| Midpoint | More accurate than left or right |
As , all Riemann sums converge to the definite integral (for continuous functions):
Error rates: Left and right sums have error . Midpoint sums have error — much faster convergence. For even better accuracy, use the trapezoidal rule or Simpson's rule.
Estimate with right Riemann sum:
Exact answer: . The right sum overestimates because is increasing.
A Riemann sum approximates the area under a curve by dividing the region into rectangles. The interval is split into subintervals, each of width , and the function is sampled at a chosen point in each subinterval.
The three main types are left, right, and midpoint Riemann sums. Left sums sample at the left endpoint of each subinterval, right sums at the right endpoint, and midpoint sums at the center. Midpoint sums are generally the most accurate.
Yes, for continuous functions, all Riemann sums converge to the definite integral as the number of rectangles approaches infinity. Left and right sums converge at rate , while midpoint sums converge faster at .
For an increasing function, the left endpoint of each subinterval is the smallest value. The rectangle height is therefore less than the actual curve height across most of the subinterval, causing the total area to be underestimated.
It depends on the function. For smooth, slowly varying functions, 10 to 50 rectangles may suffice for a rough estimate. For highly curved or oscillatory functions, you may need hundreds or thousands. Simpson's rule typically gives better accuracy with fewer subdivisions.