Visualize definite integrals as area under curves. Compare Riemann sums (left, right, midpoint, trapezoidal) with n subdivisions.
Enter a function of x, choose lower and upper bounds, and set the number of subdivisions to see a definite integral alongside left, right, midpoint, and trapezoidal approximations. The diagram shows how each rule represents the curve over equal-width subintervals, while the numerical results make overestimates and underestimates easy to compare. The most common surprise is that a definite integral is signed: regions below the x-axis contribute negative values instead of being added as positive geometric area.
A concrete input and expected output from the current implementation.
Input
f(x) = x, lower bound = 0, upper bound = 2, n = 2
Expected output
Definite integral = 2; left sum = 1; right sum = 3; midpoint sum = 2; trapezoidal sum = 2.
Each subinterval has width 1. The left heights are 0 and 1, the right heights are 1 and 2, and the midpoint heights are 0.5 and 1.5. The trapezoidal and midpoint rules both reproduce the integral of this linear function exactly.
The interval from a to b is divided into n pieces, each with width (b - a) / n. Left and right sums use function values at their named endpoints. The midpoint sum samples halfway across each piece. Each sampled height is multiplied by the common width, and those signed contributions are added.
The trapezoidal rule replaces each curved segment with a straight line joining its endpoint values. Its contribution over one subinterval is the width multiplied by the average of the two endpoint heights. Comparing its displayed shapes and value with the rectangle-based sums helps reveal how the chosen sampling rule responds to increasing, decreasing, or curved functions.
For an increasing function on each subinterval, the left endpoint gives the smaller height and the right endpoint gives the larger one, so the left sum tends to underestimate while the right sum tends to overestimate. For a decreasing function the relationship reverses. If the function changes direction, neither rule must stay entirely above or below the integral.
No, not necessarily for any finite n. More subdivisions usually reduce the error for a well-behaved function because each approximating piece covers a narrower interval. Exact agreement can occur for special functions and rules, but it should not be assumed. Convergence also depends on the function being Riemann integrable over the selected interval.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
Calculate percentages, percentage change, and what percent one number is of another with step-by-step breakdowns.
Calculate tips and split bills between multiple people with customizable tip percentages and rounding options.
Calculate monthly payments, total interest, and generate amortization schedules for loans and mortgages.
Calculate Body Mass Index from height and weight with category classification and healthy range guidance.
Calculate sale prices, savings amounts, and final costs after single or stacked percentage discounts.
Compute mean, median, mode, standard deviation, variance, and other descriptive statistics from a data set.