# Integration

In mathematics, integration is the process of finding a value for the area under a curve. It is the inverse of differentiation, which is the process of finding a value for the height of a function at a point. Integrating a function of one variable with respect to a second variable is the act of finding the area of the region "under" the graph of the function as the second variable varies. The area under the graph of a function "f"("x") (or "g"("x") or "h"("x") or even "k"("x")) is defined by the integral: This integral may be interpreted as the length of the perpendicular dropped from the end point of the interval "a" ? "x" ? "b" to the line "y" = "f"("x"). Although the integral is defined as area, this is not the visual definition of the integral. For example, the area of a rectangle with side "x" + "y" is The integral of a function "f" from "x" = 0 to "x" = "a" is the limit of the sum of the areas of the rectangle "f"("x") = 1 for "x" between "x" = 0 and "x" = "a". The integral of "f" from "x" = 0 to "x" = "a" is sometimes denoted as "f"("x") and is the value of "f"("x") at "x" = "a" or "x" ? "a". Although the area under the curve "f"("x") is the intuitive concept of the integral, the integral can also be defined by the area inside the curve, which results in a more mathematically rigorous definition. The integral of a function "f" from "x" = 0 to "x" = "a" is the limit of the sum of the areas of the parallelogram "f"("x") = 1 for "x" between "x" = 0 and "x" = "a", as "x" changes from "x" = 0 to "x" = "a". The integral of "f" from "x" = 0 to "x" = "a" is sometimes denoted as "f"("x") and is the value of "f"("x") at "x" = "a" or "x" ? "a". The integrand is a function of the variable "x" that is to be integrated, and the interval of integration "a" ? "x" ? "b". The limit of the integral is the value of the integrand at "x" = "a" or "x" ? "a". The integral can be used in both directions, i.e. as the area under the graph of the function and as the area under the curve of the function. The integral of a function "f" from "x" = 0 to "x" = "a" is the limit of the sum of the areas of the triangle "f"("x") = 1 for "x" between "x" = 0 and "x" = "a", as "x" changes from "x" = 0 to "x" = "a". The integral of "f" from "x" = 0 to "x" = "a" is sometimes denoted as "f"("x") and is the value of "f"("x") at "x" = "a" or "x" ? "a". The integrand is a function of the variable "x" that is to be integrated, and the interval of integration "a" ? "x" ? "b". The limit of the integral is the value of the integrand at "x" = "a" or "x" ? "a". A function "f" is integrable at "x" = "a" if its integral exists and is finite. The function "f" is integrable at "x" = "a" if and only if its integral exists and is equal to "f"("x") ?"x" ? "