Integral of a compound equation (or otherwise finding the area under a graph): f(x) = 10x*(x^(0.5) - 2)

This can be done 'by parts' or by expanding. In this case it would be easier to expand as it is possible to deal with terms individually here.
This becomes: 10xx^1/2 - 10x2
                      = 10x^3/2 - 20x
We know this because powers of the same value ( x in this case) work additively when multipled, so x^a*x^b = x^a+b. The other term is also just multipled as normal, two lots of 10x = 20x.
When integrating, we may remember this as the opposite of a differential, so that previously the power must have been 1 higher, and the factor is a factor of the power lower. This simply means that the integral of x^a = (1/(a+1))x^a+1. This then means that the integral with respect to x is:
(10/(5/2))x^5/2 - (20/2)x² + Constant.  This can be simplified (at this point I would show the simplificiations using either a physical whiteboard or online tools)

If we have the limits of the function, we can put those in too, remembering that we substitute it as f(upper limit) - f(lower limit) for a function of f(x). This would also find the area under a graph, if done correctly, remembering to also separate the limits based on if its above or below an axis (would need visual representation, again easy to show on a whiteboard).