How do you find the mass involved in fusion reactions in the center of a star? How have scientists found the amount of a stars mass that is in the central region where nuclear fusion occurs? For example, our Sun has about 10% of its mass in the center, how have scientists found that?
 A: The equations Countto10 is alluding to are the equations of stellar structure, which describe how the pressure ($P$), luminosity ($L$), temperature ($T$), and enclosed mass ($M$) change at a radius $r$ in the star1:
$$\frac{\mathrm{d}M_r}{\mathrm{d}r}=4\pi r^2\rho(r)\tag{Mass}$$
$$\frac{\mathrm{d}P}{\mathrm{d}r}=-\frac{GM_r\rho(r)}{r^2}\tag{Pressure}$$
$$\frac{\mathrm{d}L}{\mathrm{d}r}=4\pi r^2\epsilon(r)\tag{Luminosity}$$
$$\frac{\mathrm{d}T}{\mathrm{d}r}=\frac{3\rho(r)\kappa(r)}{64\pi r^2\sigma[T(r)]^3}L(r)\tag{Temperature}$$
$\epsilon$, $\rho$ and $\kappa$ are the rate of energy generation, density and opacity, respectively. All other symbols are constants. They're coupled differential equations, and so they must be solved by numerical methods, not analytical ones like we might use for "normal" differential equations. One of the simplest common methods is the fitting method. We know that $M(r=0)=0$ and $L(r=0)=0$, but we don't know $P(r=0)$ and $T(r=0)$. However, we can figure out the values of $P$ and $T$ at the surface.
We can guess what $L$ and $M$ will be at the surface and what $P$ and $T$ will be at the center of the star. We then "integrate" both inwards and outwards until we reach a "fitting point", and check to see whether the equations match up. If they don't, we can still make some adjustments, and redo the process. There are other tricks that make the process more interesting and certainly quicker. At the end of it all, though, you have a description of how all these variables change with radius inside the star. You can then figure out how much mass, for instance, is contained within a certain region, and if we can figure out the outer radius of the region where the primary nuclear fusion takes place, we can find $M$ at that radius. That gives you the result you're looking for.
This is all very complicated, and I probably didn't make things seem any simpler. That's okay; the process really isn't simple. Something more hands-on might convince you of the power of some of these assumptions. We can approximate the star by assuming that it is a polytrope, that is, that its pressure and density are related by a certain simple formula. After doing some mathematical playing around, we eventually arrive at a simple differential equation, called the Lane-Emden equation:
$$\frac{1}{\xi^2}\frac{\mathrm{d}}{\mathrm{d}\xi}\left(\xi^2\frac{\mathrm{d}\theta}{\mathrm{d}\xi}\right)=-\theta^n$$
Ack! More variables! $\xi$ is known as the dimensionless radius, because it scales linearly with $r$. $\theta$ is a function of $\xi$, and $P$, $\rho$ and $T$ are all functions of $\theta$. So if we solve this differential equation, we can come up with some really interesting estimates for the structure of the star!2
I had some old Python code for a fourth-order Runge-Kutta method, and I used it to numerically integrate the Lane-Emden equation (although I did not use a large step size). I then plotted the temperature, density, and pressure of the star divided by their central values:

Look at how quickly they drop off! $\theta=0$ at just over $\xi=6$, so about halfway through the star, conditions are much different than at the center. This seems to confirm the assertion that the core contains about 10% of the Sun's mass.

1 The equation for temperature is really only valid under certain conditions.

2 The assumption fails near the surface, but that's okay. We only really care about the core here.
A: We know the equations  regarding the mass/density, pressure (both gravitational inwards and radiation outwards) and temperature profile of the  structure of the sun. and these are detailed below.
We assume the simplest case of spherically symmetric quasi-static model. A quasi-static process implies that the star undergoes changes at a rate slow enough for the system to maintain internal thermodynamic equilibrium.
The Sun, and other main sequence stars, are in a balanced state between gravity acting to cause contraction and radiation pressure from the energy source at the core, as well as the gas pressure of the material of the Sun, acting to resist the inward force.
The simplest case of a large dense cloud of gas such as the Sun, requires 4 first-order differential equations: two of them describe how pressure and the hydrogen gas mass respond to changes in radius, that is how pressure and mass vary as we go from the surface of the star towards the core. The other two differential equations deal with luminosity, (the energy released by the star) and temperature and how these variables react to a reduction in radius.
We use the following variables:
Matter density ${\displaystyle \rho (r)} $
Temperature ${\displaystyle T(r)}$ 
Total pressure (matter plus radiation)  ${\displaystyle P(r)}$  luminosity ${\displaystyle l(r)}$ 
along with a variable representing energy generation rate per unit mass ${\displaystyle \epsilon (r)} $
We assume a spherical shell with a width of   ${\displaystyle {\mbox{d}}r}$ at a distance  ${\displaystyle r}$ from the center of the star.
Another simplifing assumption is that Sun is taken to conform to local thermodynamic equilibrium (LTE) and this implies that  the temperature is identical for matter and photons (radiation).
You might raise the point that LTE cannot be a good approximation, as the deeper into the Sun we get, the higher the temperature becomes, but this assumption holds because the distance over which the temperature varies is much larger than the mean free path ${\displaystyle \lambda }$ of the photons emerging from the core.  
Hydrostatic equilibrium is assumed as mentioned above: the outward force due to the pressure gradient within the star is exactly balanced by the inward force due to gravity. 
$\frac{\mathrm{d}P}{\mathrm{d}r}=-\frac{GM_r\rho(r)}{r^2}\tag{Pressure}$
where ${\displaystyle m(r)}$  is the cumulative mass inside the shell at ${\displaystyle r}$  and $G$ is the gravitational constant. 
The cumulative mass increases with radius according to the mass continuity equation:
$\frac{\mathrm{d}M_r}{\mathrm{d}r}=4\pi r^2\rho(r)\tag{Mass}$
When we integrate the above equation from $r=0$ (the center of the Sun) to $r_s$, (the somewhat arbitrary surface distance), this produces the total mass of the Sun.
Now we need to know how much energy emerges from the spherical shell.
$\frac{\mathrm{d}L}{\mathrm{d}r}=4\pi r^2\epsilon(r)\tag{Luminosity}$
     This is the energy equation.
${\displaystyle \epsilon _{\nu }}$ represents the luminosity carried, almost without any interaction on their long distance journey through the star, as neutrinos per unit mass.
Beyond the fusion region of the core, no energy is generated, so luminosity is constant in that "outer" region.
So how does the energy escape from the core, through the huge mass of gas/plasma above it?
We will ignore conductive energy transport, and deal with radiative energy transport, which corresponds to the inner region of a solar mass main sequence star.
$\frac{\mathrm{d}T}{\mathrm{d}r}=\frac{3\rho(r)\kappa(r)}{64\pi r^2\sigma[T(r)]^3}L(r)\tag{Temperature}$
where ${\displaystyle \kappa }$ is the opacity of the matter,  ${\displaystyle \sigma }$ is the Stefan-Boltzmann constant, and the Boltzmann constant is set to one.
We don't have a rigorous treatment of the third form of energy transport, i.e. convection. In the Sun, near the core, convection is adiabatic (heat does not enter or leave the region) but closer to the surface convection is not adiabatic. The mixing length theory (which can be compared analogously to the mean free path of a photon, but for pockets of fluid) contains two free parameters which must be set to make the model fit observations.
Source (for all of this answer, in one form or another) Wikipedia Stellar Structure

Any thermodynamic system requires equations of state, which link  the pressure, opacity and energy generation rate to other local variables, in the case of the Sun: temperature, density, chemical composition, etc. Relevant equations of state for pressure may have to include the perfect gas law, radiation pressure, pressure due to degenerate electrons, etc. Opacity cannot be expressed exactly by a single formula. It is calculated for various compositions at specific densities and temperatures and presented in tabular form.

Three important points preventing us from solving everything Sun related in a neat analytical form are the numerical methods used to calculate opacity and the pressure equation of state. Finally, the nuclear energy generation rate is computed from nuclear physics experiments.
All differential equations demand boundary conditions, possibly the most important of which is the values for $r$. As the pressure within the Sun is so high, it seems justifible to set the surface pressure to zero, and the temperature of the surface of the Sun is easily measured.
So from these equations, we can estimate how much of the Sun's mass conforms to the conditions necessary to initiate and maintain fusion.
Then we can check our predictions of the energy output of 10% of the Sun's mass against experimental evidence.
In theory, we could measure the neutrino output, as these pass straight through the sun, as a further check, in practice detecting neutrinos is far from easy. The fact that early experiments detected only about a third of the total expected was called the "solar neutrino problem". Hyperphysics Solar Neutrinos
