Leibniz Integral Rule - Proofs - Proof of Basic Form

Proof of Basic Form

Let:

So that, using difference quotients

Substitute equation (1) into equation (2), and combine the integrals (since the difference of two integrals equals the integral of the difference).

 u'(x) = lim_{h rightarrow 0} frac{1}{h} left = lim_{h rightarrow 0} frac{1}{h} {int_{y_0}^{y_1}left,dy}

The term is constant, so can be included in the integrand.

Provided that the limit can be passed under the integral sign, we obtain

begin{align} u'(x) &= int_{y_0}^{y_1} {partial over partial x} f(x, y),dy \ Rightarrow {dover dx} u(x) &= int_{y_0}^{y_1} {partial over partial x} f(x, y),dy
end{align}

We claim that the passage of the limit under the integral sign is valid. Indeed, the bounded convergence theorem (a corollary of the dominated convergence theorem) of real analysis states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. To complete the proof, we show that these hypotheses are satisfied by the family of difference quotients . Continuity of and compactness implies that is uniformly bounded. Uniform boundedness of the difference quotients follows from uniform boundedness of and the Mean Value Theorem, since for all y and n, there exists z in the interval such that . The difference quotients converge pointwise to since exists. This completes the proof.

For a simpler proof using Fubini's theorem, see the references.

Read more about this topic:  Leibniz Integral Rule, Proofs

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