Integral Calculus
The study of calculus can be separated into two components: derivative or differential calculus and integral calculus. Integral calculus can be thought of as dealing with the area under curves, but it also touches on two- and three-dimensional situations with path and surface integrals. On the most basic level, with finding the area under curves, integrals can be the reverse of derivatives--hence their other name, anti-derivatives. There are two major types of integrals on the lowest level: the definite and the indefinite. The definite integral provides a numerical range over which the integral is to be calculated; the indefinite integral is looking for a closed-form solution to which any range can be applied.
Integral calculus is often used in conjunction with differential equations. A differential equation will, as its name indicates, provide information about the rate of change of a function in its differential form. The process of solving the equation then usually results in an equation with an integral on each side. These one-dimensional integrals are solved with a variety of direct, numerical, and series form methods.
A change to two or three dimensions in an integral will work as if they are nested unrelated problems as long as the variable over which one integrates is not a vector. However, there are some situations in which the direction is important. Path integrals can be solved along two- or three-dimensional surfaces, but the direction of integration is important, in that the rate of change (the differential inside the integral) may vary according to how one moves from point A to point B. Therefore, a path of integration must be specified. Surface integrals generalize integration to three dimensions and deal with all three at once, so the direction of integration and the perpendicular direction are both important when calculating these integrals.
All of these major studies within integral calculus have uses in the physical world as well as in mathematical theory. The study of complex variables requires a firm basis of knowledge of integral calculus, and the physical sciences use both its anti-differential closed form and in its calculation of cumulative results or total areas in its definite form. Integral calculus provides both a test of experimental results and a way to formulate information about new theories, when it is applied to the sciences. In computer science, the numerical side of integral calculus is more important for understanding of many internal calculations, but the whole discipline is still vital for success in computer theory. In mathematics, its tenets and associated skills are assumed even when they are not directly used--and they are directly used frequently.
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