Integrals as a limit of rectangle areas

What it shows

The area between a curve y = f(x) and the x-axis on [a, b] is approximated by n rectangles (left, right, midpoint) or trapezoids; as n grows, the Riemann sum approaches the definite integral F(b) − F(a). Further screens show signed area (regions below the axis count negative, net integral versus total area), the properties ∫ₐᵇ + ∫ᵇᶜ = ∫ₐᶜ, reversed limits and linearity, the average value of f with the point c where f(c) equals it, the improper integral of 1/xᵖ from 1 to ∞, which converges only for p > 1, and a tank whose amount is the integral of the net flow rate.

How to use

Use Screen to choose a view. On Riemann sums, drag a and b, change the Number of rectangles n and compare Left, Right, Midpoint and Trapezoid; Run increases n. On the other screens, drag the limits on the graph and pick a Function; under Properties of integrals switch between Additivity, Reversed limits and Linearity. On Improper integral, set p and press Let b → ∞. On the tank screen, set Inflow and Outflow and press Run.

Parameters you can change

• Function f(x) = x², f(x) = sin x + 1, f(x) = x³ − 3x + 3, v(t) = 2t + 1 (velocity, m/s)
• Lower limit a -3–5
• Upper limit b -3–5
• Number of rectangles n 1–200
• Height sample point Left endpoint, Right endpoint, Midpoint, Trapezoid
• Screen Riemann sums, Signed area, Properties of integrals, Average value, Improper integral, Rate in – rate out (tank)

Questions to explore

1. For f(x) = x² on [0, 2], why is the left-endpoint sum always less than the integral?
2. Why does the integral of 1/x² from 1 to ∞ converge while the integral of 1/x diverges?
3. If f(x) is negative on the whole interval [a, b], what sign does the sum Sₙ have?