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.
Review Questions
What are the two main parts of the Fundamental Theorem of Calculus?
How does the first part of the theorem relate differentiation to integration?
In what way does the second part help in evaluating definite integrals?
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.