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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.16 Differential Forms and Integration

 Terence Tao

It goes without saying that integration is one of the fundamental concepts of single-variable calculus. However, there are in fact three concepts of integration that appear in the subject: the indefinite integralf (also known as the antiderivctive), the unsigned definite integral ∫[ab] f (x) dx (which one would use to find the area under a curve, or the mass of a one-dimensional object of varying density), and the signed definite integral Image f(x) dx (which one would use, for instance, to compute the work required to move a particle from a to b). For simplicity we shall restrict our attention ...

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