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Unformatted text preview: Math 185. Differentiating under the Integral We begin with Fubinis theorem : Theorem (Fubini). Let a < b and c < d be real numbers, and let g : [ a,b ] [ c,d ] R be a continuous function. Then Z b a Z d c g ( x,y ) dy dx = Z d c Z b a g ( x,y ) dxdy . (In more general settings, such as those involving improper integrals, discontinuous functions, or more advanced concepts of integration, the theorem would assume that g is absolutely integrable: R b a R d c | g ( x,y ) | dy dx < . In the situation here, though, this condition follows from continuity of g and compactness of its domain.) This allows you to differentiate real integrals under the integral sign: Corollary. Let I be an open interval, and let : [ a,b ] I R be a continuous function. Assume that is differentiable in its second variable y , and that y is continuous on [ a,b ] I . Let : I R be the function given by ( y ) = Z b a ( t,y ) dt, y I ....
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.
- Fall '07
- Real Numbers