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Back to Integral Calculus

Fundamental Theorem of Calculus

Sweep an area along the axis and plot the running total. Its slope at every x is exactly f(x): integration and differentiation undo each other.

Step 1 of 16

— f(t)— A(x), the area so far

Speed

Top: f(t) = (t − 1)(t − 3). Bottom: A(x), the signed area under f from 0 to x. Slide x to the right and keep a running total of the area swept so far.

A(x) = \int_{0}^{x} (t - 1)(t - 3)\,dt, \qquad A(0) = 0

\frac{d}{dx}\int_{a}^{x} f(t)\,dt = f(x) \qquad\text{and}\qquad \int_a^b f(t)\,dt = F(b) - F(a)

Fundamental Theorem of Calculus

Sweep an area along the axis and plot the running total. Its slope at every x is exactly f(x): integration and differentiation undo each other.

Step 1 of 16

— f(t)— A(x), the area so far

Speed

Top: f(t) = (t − 1)(t − 3). Bottom: A(x), the signed area under f from 0 to x. Slide x to the right and keep a running total of the area swept so far.

A(x) = \int_{0}^{x} (t - 1)(t - 3)\,dt, \qquad A(0) = 0

\frac{d}{dx}\int_{a}^{x} f(t)\,dt = f(x) \qquad\text{and}\qquad \int_a^b f(t)\,dt = F(b) - F(a)

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Fundamental Theorem of Calculus

See also

Fundamental Theorem of Calculus

See also