Definitions of integrals

Definitions of integrals - Mathematics 20E Spring 2009...

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Unformatted text preview: Mathematics 20E - Spring 2009 - Integrals Ryan Szypowski May 17, 2009 The following is a short set of notes reviewing the types of integrals dis- cussed in Mathematics 20E. These notes are not meant to be a complete description of the integration process, but just as a tool to organize the ma- terial. 1 Basic Integrals 1.1 One Dimension Let f : [ a, b ] ⊂ Ê → Ê be a continuous function. The integral can be defined as integraldisplay [ a,b ] f dx = integraldisplay b a f ( x ) dx where the right-hand integral can be described in terms of Riemann sums. Generally, the Fundamental Theorem of Calculus is used to compute this as integraldisplay b a f ( x ) dx = F ( b ) − F ( a ) where F : [ a, b ] ⊂ Ê → Ê is any function such that F ′ ( x ) = f ( x ). This integral can be interpreted in a couple of ways. It can represent the signed area between the graph y = f ( x ) and the x-axis, or if we think of f as a density in units of mass per length, then the integral is the mass of the interval [ a, b ]. 1 1.2 Two Dimensions Let f : D ⊂ Ê 2 → Ê be a continuous scalar field. The integral integraldisplayintegraldisplay D f dA can be defined directly in terms of Riemann sums, but we generally compute it in one of two ways depending on the region D . If the region D can be described by D = { ( x, y ) : a ≤ x ≤ b, l ( x ) ≤ y ≤ u ( x ) } for some constants a, b and functions l, u : [ a, b ] ⊂ Ê → Ê , then we can use Fubini’s theorem to iterate the integral as a pair of nested one dimensional integrals, integraldisplayintegraldisplay D f dA = integraldisplay b a parenleftBigg...
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Definitions of integrals - Mathematics 20E Spring 2009...

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