Calculus

The Fundamental Theorem

Sweep a point across a curve and watch the area behind it grow into a new function, whose slope at every point is the height of the curve.

f > 0: A climbsf < 0: A fallsheight f(x) = slope of AA(b) − A(a)

Drag across either plot to move x

Area so far, A(x)—

—

Function

Point x 1.80

Start a 0.00

Riemann pieces

Method

Pieces 12

Values

f(x)—

A(x)—

∫_a^b f = A(b) − A(a)—

Riemann sum—

error of the sum—

Error against pieces, log–log

leftrightmidtrap

What to observe

Sweep x: A climbs fastest where f is tallest. The amber bar (the height f(x)) and the amber tangent on A (its slope) are the same number.
On x³ − x the slivers below the axis count as negative, so A runs downhill there, and A is flat exactly where f crosses zero.
Move the start a: A drops by a constant C but keeps its shape, so both curves rise by the same amount from a to b. Any antiderivative F gives ∫ₐᵇ f = F(b) − F(a).
With Left pieces, doubling n only halves the error, while Midpoint quarters it: in the log–log plot Left and Right fall with slope −1, Midpoint and Trapezoid with slope −2.

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