The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. If f(x) is continuous over an interval [a, b], then there is at least one point c ∈ [a, b] such that f(c) = 1 b − a∫b af(x)dx. then F′ (x) = f(x).What is the fundamental theorem of line integrals?
Theorem 16.3.1 (Fundamental Theorem of Line Integrals) Suppose a curve C is given by the vector function r ( t), with a = r ( a) and b = r ( b). Then provided that r is sufficiently nice. Proof. We write r = ⟨ x ( t), y ( t), z ( t) ⟩, so that r ′ = ⟨ x ′ ( t), y ′ ( t), z ′ ( t) ⟩.What is the mean value theorem for integrals?
The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value c such that f(c) equals the average value of the function. The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral.What is the first part of the first fundamental theorem?
The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions.