Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of , related to differentiation by the fundamental theorem of calculus:

If $f$ is the derivative of $F$, then $$\int^b_a f(x)\text{ }\mathrm dx = F(b)-F(a)$$

Note: to be strict, the equality in the formula above is not always achieved, but requires $f$ to have some good property, for example, $f$ is absolutely continuous and $f$ is the derivative of $F$ almost everywhere. However, for most functions encountered in real-world problems, the equality in above formula is true.

Integration is often harder than differentiation. Several techniques have been developed, including integration by parts, integration by substitution, trigonometric substitutions and partial fractions.

Integration can be used to find the area under a graph and finding the average of the function. It can also be used to compute the volume of certain solids.

There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals

history | excerpt history