order of integrals

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CALCULUS

Topic : Changing Order of Integrals

By : Taher K D

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Integrals

The integral is an important concept in mathematics.

The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by

The founders of calculus thought of the integral as an infinite sum of rectangles of infinitesimal width.

Darboux upper sums of the function y = x2

Darboux lower sums of the

function y = x2

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Multiple Integral

The multiple integral is a generalization of the definite integral to functions of more than one real variable, for example, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals.

Integral as area between two curves.

Double integral as volume under a surface

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Changing Order of integration

In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.

The problem is evaluation of an integral of the form

Integration over the triangular area can be done using vertical or horizontal strips as the first step. This is an overhead view, looking down the z-axis onto the x-y plane. The sloped line is the curve y = x.

If f(x,y) is a continuous function on the rectangle [a,b]*[c,d] then

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