The Fundamental Theorem of Calculus | Connecting Integration and Differentiation for Definite Integrals

Fundamental Theorem of Calculus: “If f is continuous on [a,b] and F is an antiderivative of f on [a,b], then…”

If f is continuous on the interval [a,b] and F is an antiderivative of f on [a,b], then the Fundamental Theorem of Calculus states that:

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If f is continuous on the interval [a,b] and F is an antiderivative of f on [a,b], then the Fundamental Theorem of Calculus states that:

1. The definite integral of f(x) from a to b is equal to F(b) – F(a).
In other words, if we want to find the area under the curve of the function f(x) between x=a and x=b, we can evaluate the antiderivative F(x) at the endpoints and subtract the values.

This result establishes a deep connection between the concepts of integration and differentiation, and it has two parts:

2. First Part (also known as the evaluation part): The theorem guarantees that if we have a continuous function f(x) on [a,b], there exists an antiderivative F(x) of f(x). This means that we can find a function whose derivative is equal to f(x).

3. Second Part (also known as the accumulation part): The theorem states that the definite integral of f(x) on [a,b] can be evaluated by finding an antiderivative F(x) of f(x) and subtracting the values of F(x) evaluated at the endpoints a and b. This provides a way to calculate the exact value of the area under the curve.

In summary, the Fundamental Theorem of Calculus connects the concept of finding antiderivatives (integration) with the concept of finding areas under curves (definite integration). It allows us to evaluate definite integrals by finding antiderivatives and applying the endpoint values.

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