I want to make a graph of the integral of a function (on the y-axis) where its integration is with respect to one of its variables, with respect to another one of its variables (on the x-axis). Please pardon me, the previous sentence is rather confusing because I am pretty confused myself; this is my first time dealing with more than two variables.
Here are the specifics: I want to get the area under the curve of the Maxwell-Boltzmann distribution for chemical reactions and see how that area changes as absolute temperature, another one of its variables changes. The equation is given below:
$f_E(E) = \left(\frac{1}{\pi k_BT}\right)^{3/2} 2\pi \cdot E^{1/2} e^{-\frac{E}{k_B T}}$
I want to see how the integration of the above equation changes with temperature. how this
$f_E(T) =\int_{E_a}^{\infty } \left(\frac{1}{\pi k_BT}\right)^{3/2} 2\pi \cdot E^{1/2} e^{-\frac{E}{k_B T}}dE$
changes with T, temperature.
With my limited exposure and knowledge, I've tried a number of 3D graphing software including grapher and geogebra but I've only gotten as far as making a 3D graph of the normal distribution curve versus temperature, not its integral. For context, my current lesson in high school are derivatives of trigonometric functions.
Any help would be appreciated.
Edit: (for graphing the integral vs. T on Desmos) I graphed the following, where T = x.
$ f(x)=\frac{m}{kx}\left(1-erf \left(\frac{E_a \sqrt{m}}{\sqrt{2kx}}\right) \right) $
Where k is the Boltzmann constant, $1.38064852 × 10^{-23}$
$m = 4.022 \times 10^{-25}$
$E_a = 101kJmol^{-1}$
I got the equation from here https://physics.stackexchange.com/questions/366804/kinetic-energy-in-maxwell-boltzmann-distribution and https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution
This coming from statistical thermodynamics, it should be $n(T)$ (as already said in comments).
$$n(T)=4\pi\left(\frac{m}{2\pi kT}\right)^{\frac32}\int_{E_a}^{\infty } e^{-\frac{mx^2}{2kT}}dx$$ and we face a gaussian integral. So, $$\int_{E_a}^{\infty} e^{-\frac{mx^2}{2kT}}dx= \sqrt{\frac{\pi k T}{2m}}\,\text{erfc}\left(\frac{E_a \sqrt{m}}{ \sqrt{2k T}}\right)$$ $$n(T)=\frac{m}{k T}\,\text{erfc}\left(\frac{E_a \sqrt{m}}{ \sqrt{2k T}}\right)$$ Compute the deivative $n'(T)$