The Fundamental Theorem of Calculus links the concept of differentiation and integration. It states that if a function is continuous over an interval, then its integral can be computed using its antiderivative.
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The Fundamental Theorem of Calculus has two parts: the first part relates the derivative to the integral, and the second part provides a way to evaluate definite integrals.
The first part states that if $F$ is an antiderivative of $f$ on an interval $[a, b]$, then for every $x$ in that interval, $\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$.
The second part states that if $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.
This theorem confirms that differentiation and integration are inverse processes.
Understanding both parts is crucial for solving problems involving definite integrals and their applications.
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Related terms
Antiderivative: A function whose derivative is the given function.
Definite Integral: The evaluation of an integral within specific limits, providing a numerical value.