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| Multiple definite integrals are generally expressed with a set notation. For example, consider a region ''D'' defined as the area between two functions: ''f(x)'' and ''g(x)''. The region is expressed as ''D = {(x,y) | a ≤ x ≤ b, f(x) ≤ y ≤ g(x)}''. | 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'''). |
Integral
An integral is the inverse of a derivative; an antiderivative.
Description
A definite integral represents the area under a curve from a to b. It is notated as:
Fundamental to solving such integration problems is understanding that integrals are antiderivatives. Reference 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:
An improper integral sets an infinite bound, like:
Multiple Integrals
When integrating with respect to multiple variables, the order of integration can be swapped. For this reason, a shorthand has emerged:
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:
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:
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:
Vector-Valued Functions
The indefinite integral of a vector-valued function is given by:
Letting F, G, and H be the antiderivatives of f, g, and h respectively; and letting C be a vector of Ci, Cj, and Ck, this integral can be expressed as:
