1st Fundamental Theorem of Calculus
the integral of f(x) from a to b = f(b) – f(a)
The 1st Fundamental Theorem of Calculus is a fundamental concept in calculus that demonstrates the relationship between differentiation and integration.
The theorem states that if f(x) is continuous on [a, b], and F(x) is an antiderivative of f(x) on [a, b], then:
∫a^b f(x) dx = F(b) – F(a)
This means that the definite integral of a function f(x) over an interval [a, b] can be evaluated by finding an antiderivative (or indefinite integral) of the function F(x), evaluating it at b and a, and then subtracting those values.
In essence, the theorem tells us that differentiating an antiderivative gives us back the original function that we started with; and integrating the original function gives us back the antiderivative.
This theorem is a crucial tool in calculus, as it simplifies many problems where we need to evaluate integrals. It also establishes a connection between two of the most important operations in calculus, leading to the development of the Fundamental Theorem of Calculus, which involves both differentiation and integration.
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