= Integral =

An '''integral''' is the inverse of a [[Calculus/Derivative|derivative]]; an '''antiderivative'''.

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== Description ==

A '''definite integral''' represents the area under a curve from ''a'' to ''b''. It is notated as:

{{attachment:def.svg}}

Fundamental to solving such integration problems is understanding that integrals are '''antiderivatives'''. Reference [[Calculus/Derivative#Rules|derivative rules]] and apply them in inverse. Concretely, letting ''F'' be the antiderivative of ''f'', and assuming ''f'' is integrable on ''[a,b]'', the above can be solved as:

{{attachment:theory2.svg}}

An '''improper integral''' sets an infinite bound, like:

{{attachment:imp.svg}}



=== Multiple Integrals ===

When integrating with respect to multiple variables, the order of integration can be swapped. For this reason, a shorthand has emerged:

{{attachment:multi.svg}}

where ''D'' represents a region. This region is then separately expressed with a set notation. For example, consider a region ''D'' defined by values of ''x'' between ''a'' and ''b'' and values of ''y'' between ''c'' and ''d''. The region is expressed as ''D = {(x,y) | a ≤ x ≤ b, c ≤ y ≤ d}''.

The solution to this integral is more clear in the '''iterated integral''' form:

{{attachment:iter.svg}}

That is, integrate with respect to one variable while holding the other constant, then integrate with respect to the other. This holds whenever ''f'' is integrable over ''D'' ('''Fubini's theorem''').



=== Indefinite Integrals ===

An '''indefinite integral''' is the generalization of integration, notated as:

{{attachment:indef1.svg}}

Because the derivative of a constant is 0, there are infinitely many antiderivatives for any given function. To represent the generalized set of antiderivatives, add a constant ''C''.

Letting ''F'' be the antiderivative of ''f'', an indefinite integral is solved as:

{{attachment:indef2.svg}}



=== Vector-Valued Functions ===

The indefinite integral of a vector-valued function is given by:

{{attachment:vec1.svg}}

Letting ''F'', ''G'', and ''H'' be the antiderivatives of ''f'', ''g'', and ''h'' respectively; and letting ''C'' be a vector of ''C,,i,,'', ''C,,j,,'', and ''C,,k,,'', this integral can be expressed as:

{{attachment:vec2.svg}}






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