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Is it possible to derive analytically the mean value of the sine wave function (expressed as $f(x) = mean + A*\sin(x)$) based on known area under the curve and amplitude ($A$) (for illustration of what I mean please see below. Or it has only a numerical solution? Thank you for any suggestions.

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    Why ''functional analysis'' and ''analyticity'' as tags? – 2017-02-14

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I will use $k$ for the mean value, so $f(x)=k+A\sin(x)$ and $V$ for the area. The curve passes through $0$ when $x=\arcsin (-\frac kA)$ so we have $$V=2\int_{\arcsin(-\frac kA)}^{\frac \pi 2} (k+A\sin x) \ dx\\ =2(kx|_{\arcsin(-\frac kA)}^{\frac \pi 2})-2(A \cos x)|_{\arcsin(-\frac kA)}^{\frac \pi 2}\\=k\pi-2k\arcsin(-\frac kA)-2\sqrt{1-\frac{k^2}{A^2}}$$ and the mix of inverse trig and the square root make me think a numerical solution is in your future.

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    Thank you very much for your quick analysis and response. I thought that a numerical solution will be inevitable in this case. Hopefully, I will be able to solve it. – 2017-02-14