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单词 DerivativeOfRiemannIntegral
释义

derivative of Riemann integral


Let f be a continuous functionMathworldPlanetmathPlanetmath from an open subset A of 2 to .  Suppose that also the partial derivativeMathworldPlanetmathft(x,t)  is continuous in A which contains the line segments along which the integration is performed and that a(t) and b(t) are real functions differentiableMathworldPlanetmathPlanetmath in some point t0.  Denote

F(t)=a(t)b(t)f(x,t)𝑑x

and

G(t)=b(t0)f(b(t),t)-a(t0)f(a(t),t)+a(t)b(t)ft(x,t)𝑑x.

Then one has the derivativePlanetmathPlanetmath

F(t0)=G(t0)

in all such points  t=t0.

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更新时间:2025/5/5 0:55:32