Plot3D of the mean value of a function

Question

I have the following function:

$$f(x,j,G) = -(6.699719\times10^{-6})\times j\times G\times \frac{\sinh(\frac{x}{G})}{ \cosh(\frac{L}{2G})} $$

where $L=100\times10^{-6}$. I'm trying to plot its average value over $-L/2\leq x\leq L/2$ versus some values of $j$ and $G$. This is my attempt but does seem to be correct:

Plot3D[NIntegrate[Abs[-(6.699719*^-6) j G Sinh[x/G]/Cosh[100*^-6/(2 G)]], {x, -50*^-6,50*^-6}]/((50*^-6) - (-50*^-6)), {j, 1*^9, 1*^11}, {G, 1*^-6,10*^-6}]

I get different values in R. Any effort is appreciated.

Answer 1

The function $f$ as written at the top of your question is odd in x (basically you're just integrating Sinh), so your mean $f$ vanishes integrating over $-\frac{L}{2} \le x\le \frac{L}{2}$. Everything else is along for the ride.

The numerical expression you're plotting is averaging Abs[Sinh[x]] which is even. That's fine and can still be done analytically.

In essence your calculation looks like:

meanAbsSinhX[L_, G_] = Integrate[Sqrt[Sinh[x/G]^2], {x, -L/2, L/2}]/L;

meanf[const_, G_, j_, L_] :=  Abs[const j G /Cosh[L/(2 G)]] meanAbsSinhX[L, G]

We can compare this with numerical integration on human readable values:

 meanf[1, 1, 1, 1] // N

0.226362

NIntegrate[ Abs[Sinh[x]/Cosh[1/2]], {x, -1/2, 1/2}]

0.226362

And you can reproduce your above plot with:

Plot3D[meanf[-(6.699719*10^-6) , g, j, 100*10^-6] // Evaluate, {j, 
  1.*10^9, 1.*10^11}, {g, 1.*10^-6, 10.*10^-6},ColorFunction -> None]

I should point out -- I have absolutely no way of knowing whether your numbers make sense, or if this is even the value you want to be plotting. The good news is your expression isn't very complicated and can be debugged analytically.