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The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations can be ...
The sum rule (+) ′ = ′ + ′ The ... This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus ...
For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. [1] The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory . [ 2 ]
Simplest rules Sum rule in integration; Constant factor rule in integration; Linearity of integration; Arbitrary constant of integration; Cavalieri's quadrature formula; Fundamental theorem of calculus; Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the ...
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where () = is constant, () =, and (,) = does not depend on .
Fubini's theorem; Fundamental theorem of calculus; G. General Leibniz rule; Generalized Stokes theorem; ... Intermediate value theorem; Inverse function rule;
The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. If φ : U ⊆ R n → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point q, then
In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to reproduce other mathematical theories including vector calculus , differential geometry , and differential forms .