When the integral refuses to be exact
An engineer has sensor readings of water flow taken every minute and needs the total volume that passed. There is no formula for the flow, only scattered data points, so no antiderivative exists to evaluate. The trapezoidal rule handles exactly this situation: it estimates the area under the curve by slicing it into trapezoids, and this calculator runs the whole computation.
The formula, in plain terms
Formula: T is approximately h/2 x [f(x0) + 2 x (f(x1) + ... + f(xn-1)) + f(xn)], with h = (b - a)/n
Divide the interval [a, b] into n equal strips of width h. On each strip, replace the curve with a straight line between its endpoints, turning the slice into a trapezoid. The formula averages the left and right heights of every strip, which is why the interior points get doubled: each one serves as the right edge of one trapezoid and the left edge of the next, while the two endpoints are counted once.
How to use the calculator
- Enter the function, or supply your data points directly.
- Enter the interval bounds a and b.
- Choose n, the number of strips; more strips means a tighter estimate.
- Read the estimate and, if shown, the rough error bound.
Worked example: integral of x^2 from 0 to 2, with n = 2
Here h = (2 - 0)/2 = 1, and the sample points are x0 = 0, x1 = 1, x2 = 2. The function values are f(0) = 0, f(1) = 1, f(2) = 4. Plug into the formula:
- T is approximately (1/2) x [0 + 2 x 1 + 4] = (1/2) x 6 = 3.
The exact integral is 8/3, about 2.667, so the estimate overshoots by roughly 0.333. The overshoot is no accident: x^2 curves upward, so straight-line tops sit above the curve on every strip. Doubling n to 4 would shrink the error to about a quarter of this, the typical behavior for smooth functions.
Where the trapezoidal rule turns up
Any time data arrives as measurements rather than formulas, numerical integration is the only option: total rainfall from hourly gauges, distance from sampled speeds, energy from power readings. When an exact antiderivative does exist, the area under the curve calculator gives the precise answer to compare against, and the Fourier transform calculator applies similar discrete-sum thinking to frequency analysis.
Mistakes to avoid
- Using too few strips on a wiggly function. Rapid oscillation between sample points is invisible to the rule; increase n until the answer stabilizes.
- Forgetting to double the interior points. The most common hand-calculation error is treating every height equally; the endpoints are special and counted once.
- Expecting exactness. The rule is an approximation by design. For straight-line functions it is exact, for everything else it carries error that only shrinks with more strips.
What to keep in mind
The rule systematically overestimates upward-curving functions and underestimates downward-curving ones, so knowing the curve's shape tells you which side of the truth your answer sits on. For very rough or spiky data, consider Simpson's rule or simply more strips before trusting the number in a decision.