integral

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 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 infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.
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. R

    Help Integral tulooong!

    Kaka start palang ng integral namin, binigyan na agad kami nito. Na search at na sagot ko yung iba. Pero ito talaga dalawa kasi 'dx'. Wala ako ma kita paano gawin. Di naman na cacancel using u substitution. Patulong po sa mga geniuses jan,.
  2. K

    Help Math HELP!

    Hi guys! Di ko magets mgananseasearch ko sa internet. Why do we need to add "C" in indefinite integral? Like why is is it needed in solving indefinite integral calculus? Please send help!! Di ko talaga ma gets yung mga english.
  3. K

    Pubg ping problem using vps

    VPS speedtest: Download speed: 30mbps Upload speed: 10mbps Latency: 67ms While in pubg i have 360+ ms and the game can not be played with my vps.Why does this happen?
  4. K

    Calculus! integral!

    Dito na ako nagpost para maraming makapansin at sumagot agd kung alam nila.. sa April 11 pa naman ang pasahan neto, haha,. take home final examination namin.. galing e, ahaha.. so please help me, hehe.. inupload ko nlang kase ayaw basahin yung ibang symbols e..
  5. K

    Help. calculus problem..

    I know Im on the wrong path of posting this kind of question.. hihi. ito kase yung palaging may tao .hehe, baka mapansin, patulong sana ako. Yung mga mgaling jan sa Math especially calculus. thanks! kelangan ko na bukas to. huhu. :(
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