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Unformatted text preview: Math 185. Differentiating under the Integral We begin with Fubini’s 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 φ : [ a,b ] × I → R be a continuous function. Assume that φ is differentiable in its second variable y , and that φ y is continuous on [ a,b ] × I . Let ψ : I → R be the function given by ψ ( y ) = Z b a φ ( t,y ) dt, y ∈ I ....
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 Fall '07
 Lim
 Real Numbers, Derivative, Continuous function, dt

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