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 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)}.
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:
