Question

. State and prove mean value theorem for R-S integral.
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Answer 1

Step-by-step explanation:

Theorem 1 (The Second Mean-Value Theorem for Riemann-Stieltjes Integrals): Let be an increasing functionon and let be continuous on . Then there exists a point such that $\displaystyle{\int_a^b f(x) \: d \alpha (x) = f(a) \int_a^{x_0} d \alpha(x) + f(b) \int_{x_0}^b \: d \alpha (x)}$.

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Answer 2

Answer:

Theorem 1 (The Second Mean-Value Theorem for Riemann-Stieltjes Integrals): Let be an increasing function on and let be continuous on . Then there exists a point such that $\displaystyle{\int_a^b f(x) \: d \alpha (x) = f(a) \int_a^{x_0} d \alpha(x) + f(b) \int_{x_0}^b \: d \alpha (x)}$.