Integral calculus Definition and 154 Threads

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
The integrals enumerated here are those termed definite integrals, which can be interpreted formally as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

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  1. phoenixthoth

    Integral Calculus Bee: Antidifferentiating & Finding Primative Functions

    And, yes, I mean what you think it means: antidifferentiating. Finding a primative. Whatever... At my school, we're conducting an integration bee not unlike similar bees done elsewhere. The purpose of this thread is two-fold: 1. To compile a list of non-routine integrals 2. To discuss...
  2. JasonJo

    Head scratching integral calculus hw

    hey guys, i need help on my integral calculus hw 2) int(x*3*e^(x^7)) dx this one is just giving me insane problems
  3. F

    How Do You Calculate Acceleration and Angular Speed in Physics Problems?

    1. What constant negative acceleration will enable a driver to decrease the speed from 120 km/hr to 60 km/hr while traveling a distance of 100m? Where I got stuck [WIGS]: Let us say you don't know the formula used in kinematics... the acceleration is unknown. Let acceleration be a... so...
  4. F

    Get Help on Integral Calculus: Velocity & Distance Equation

    I need guidance (updated) The website has changed a lot! Very beautiful and attactive! I like it. Well, I need help. This is on integral calculus :rolleyes: . How do I find an equation involving the velocity "v" and the distance "s" given that the acceleration "a" is 800; and that v =...
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