Calculus

:For other uses of the term calculus see calculus (disambiguation)

Integral calculus

Main article: Integral

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The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula

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:mathrm{Distance} = mathrm{Speed} cdot mathrm{Time}

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for calculating the distance a car moves during a period of time when it is traveling at constant speed. The distance moved is the cumulative effect of the small distances moved in each of the many seconds the car is on the road. The calculus is able to deal with the natural situation in which the car moves with changing speed.

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Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact distance.

Related Topics:
Approximation - Riemann sum

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More formally, we say that the definite integral of a function on an interval is a

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limit of Riemann sum approximations.

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Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of the solutions of many, many smaller problems.

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The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many very tiny squares and adding the areas of those squares. (If the region has a curved boundary, then omitting the squares overlapping the edge does not cause too great an error.) Surface areas and volumes can also be expressed as definite integrals.

Related Topics:
Square - Surface area

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Many of the functions that are integrated are rates, such as a speed. An integral of a rate of change of a quantity on an interval of time tells how much that quantity changes during that time period. It makes sense that if one knows their speed at every instant in time for an hour (i.e. they have an equation that relates their speed and time), then they should be able to figure out how far they go during that hour. The definite integral of their speed presents a method for doing so.

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Many of the functions that are integrated represent densities. If, for example, the pollution density along a river (tons per mile) is known in relation to the position, then the integral of that density can determine how much pollution there is in the whole length of the river.

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Probability, the basis for statistics, provides one of the most important applications of integral calculus.

Related Topics:
Probability - Statistics

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