Area Under Curve Calculator

This calculator approximates the area under the fixed curve f(x) = x² between a Lower and Upper Bound, letting you compare three numerical integration techniques — the trapezoidal rule, Simpson's rule, and the midpoint rule — against the exact analytic answer, (b³ minus a³) divided by 3. Of the boundary and interval inputs, the Upper Bound carries far more weight over the reported area than Number of Intervals does: because x² is a smooth, well-behaved curve, moving from 100 subdivisions to 90 or 110 changes the trapezoidal answer by less than the fifth decimal place the page actually displays, so the visible result barely shifts. That near-zero sensitivity is a Trapezoidal-rule fact, not a universal one — the Midpoint Rule has its own, differently-sized O(h²) error term for this curve, and at these bounds it is just barely large enough that the same 90-vs-110 nudge can flip the last displayed digit of Number of Intervals, even though the Trapezoidal default does not. The reported Error Estimate is not measured from the approximation itself — it's a textbook formula with the known second or fourth derivative of x² already plugged in.

For Simpson's Rule specifically, that formula multiplies by the curve's fourth derivative, and since a quadratic's fourth derivative is exactly zero, the Simpson's error readout always shows exactly 0 by design, regardless of how few intervals are chosen. That's not just a display quirk here: Simpson's Rule is mathematically exact for any polynomial up to degree 3, so for this fixed x² curve the readout of 0 is the mathematically correct answer, not an approximation of one — at the default bounds and interval count, the computed area and the exact area agree to all five displayed decimal places (0.33333). That exactness is a property of this specific curve, though, not of Simpson's Rule in general: it would stop being exact for a higher-degree or non-polynomial curve, which this calculator doesn't support since f(x) = x² is fixed.