Double and triple integral issues

I'm having a really hard time trying to figure out how to find the limits of double and triple integrals.

I'm also having trouble determining which limits go on which integral and whether it should be ordered $\displaystyle dydx$ or $\displaystyle dxdy$, etc.

Here's a few problems from the book:

Find the area of the region R that lies below the parabola y=4x-x^2 above the x-axis, and above the line y=-3x+6.

It shows that area = $\displaystyle \int_{1}^{2} \int_{-3x+6}^{4x-x^2} dydx + \int_{2}^{4} \int_{0}^{4x-x^2} dydx$ (I'm not sure what's wrong with the LaTeX? here's what I used: \int_{1}^{2} \int_{-3x+6}^{4x-x^2} dydx + \int_{2}^{4} \int_{0}^{4x-x^2} dydx )

No idea how they got those limits.