Antidifferentiation, or integration
, is the reverse of differentiation
, just as multiplication
is the reverse of division
. Just as when one differentiates a function
one obtains a derivative
, when one antidifferentiates a function one obtains an antiderivative
. Differentiation and antidifferentiation are the whole point of calculus
There are two types of antidifferentiation: definite
. The definite integral from a
of some continuous function f'(x) (where a and b are constants in the domain of f'(x) and f'(x) is continuous over the interval from a to b and f'(x) exists over the interval from a to b) is approximated
by a Riemann sum
over the interval
a <= x <= b. The definite integral over that same interval is exactly equal to the limit of that same Riemann sum as the number of partitions goes to infinity. Or, in other words, as one slices the area trapped between the curve and the horizontal axis into thinner and thinner rectangles, the sum of the area of those rectangles approaches the area trapped between the curve and the horizontal axis. The fundamental theorem of calculus states that this value is exactly equal to f'(b) - f'(a).
meaning of the antiderivative is the area trapped between a curve
and the horizontal axis
. However, because the definition of an integral
derives from taking the limit of the sum of a bunch of signed quantities (rectangles whose area is the product of the value of the function being antidifferentiated at some point and a small delta, which may be positive or negative), the definite integral (or definite antidifferential) of a function may be negative, whereas in geometry area is understood to be positive.
The indefinite integral represents the whole family of functions whose derivatives result in the integrand
There are many rules for antidifferentiation, but an explanation of these rules is equivalent to a calculus course.