Interchange integrals, definition

II. The interchange of the limits of a definite integral causes the integral to change its sign. It is evident that... 

The first two integrals can each be simplified with the help of the mean value theorem let/(x, b) be the mean of the first integrand and fix, a) be the mean of the second. The third integral can be simplified by interchanging the limit with the integral and applying the definition of a derivative. Hence, we are left with... 

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Using the definitions of the norms (12.449) and (12.450) and further noting that the order of the differentiation and integration operations may be interchanged as the integral limit is not a function of e, the latter expression can be given as ... 

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