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The Second Mean Value Theorem in the Integral Calculus

Published online by Cambridge University Press:  24 October 2008

A. C. Dixon
A. C. Dixon
Affiliation:
Trinity College
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Extract

Hobson has given a proof of this theorem in its fullest generality. The present note gives an alternative for part of Hobson's argument. The theorem may be stated in two forms. If f(x) is a function of x, monotone when a ≤ x ≤ b, and φ(x) is integrable over the same range, then

where a ≤ X ≤ b,

(ii) the same holds with a < X < b except in some trivial cases where f(x) is constant in the open interval a < x < b. The form (ii) is not mentioned by Hobson.

Type
Articles
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 3 , July 1929 , pp. 282 - 284
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* Proc. Lond. Math. Soc., Ser. 2, Vol. VII, pp. 14–23 (1909).Google Scholar

* For the method of proof of this theorem, see Phil. Trans. Roy. Soc. A, 211, pp. 413–416 (1911).Google Scholar