Integral calculus theorems
The Fundamental Theorem of Calculus and the theorems built on it.
Fundamental theorem of calculus, part 1
\frac{d}{dx} \int_a^x f(t)\,dt = f(x)
FTC part 1: if f is continuous, the derivative of the integral from a to x of f(t) dt is f(x). Statement, graph, chain rule version and worked examples.
Fundamental theorem of calculus, part 2
\int_a^b f(x)\,dx = F(b) - F(a)
FTC part 2: the definite integral of f from a to b equals F(b) − F(a) for any antiderivative F. Statement, graph, a key counterexample and worked examples.
Mean value theorem for integrals
f(c) = \frac{1}{b - a} \int_a^b f(x)\,dx
The mean value theorem for integrals: a continuous function on [a, b] equals its average value at some point c. Statement, graph, counterexample, examples.
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