Definite integrals, areas and volumes

Definite integrals, areas and volumes · Diagrams

From the source curve to left and right rectangle sums

Compare left and right rectangle sums

Illustrative curve f(x) = 2 + x − x² on [0, 1.5]. Change the strip count and endpoint rule; the source drawing supplies a rising-then-falling shape, not this formula.

Teaching and source notes

The original shows the area A, then left heights f(xᵢ) and right heights f(xᵢ₊₁), with equal width Δx = (b − a)/n. The explorer keeps those choices and adds a midpoint comparison using the existing workbench.

The chosen numerical example rises and then falls. Its exact integral is [2x + x²/2 − x³/3] from 0 to 1.5, which is 3. The formula, bounds and value are illustrative, not extra givens in the source.

On a nonmonotone curve, the endpoint name alone does not prove that a sum is below or above the integral. Inspect the whole interval. Increasing n numerically illustrates convergence; it does not prove the general theorem.

Three source figures reconstructed: area A under a continuous nonnegative curve from a to b, left rectangles of height f(x_i), and right rectangles of height f(x_(i+1)). The curve rises then falls, so endpoint choice alone gives no global under/over estimate. All widths are delta x=(b−a)/n.

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