{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ho2 - Math 185 Differentiating under the Integral We begin...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

Page1 / 2

ho2 - Math 185 Differentiating under the Integral We begin...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online