Area Between Three Curves With Two Similar Lines

Question

I have a little Calculus problem which is confusing me quite a bit, so I thought to ask you guys for help. The problem consists in calculating the area between three curves, they are:

$$ - y = x² - 6x + 8 $$ $$- y = 2x - 4 $$ $$ - y = x + 2 $$

Here's the graph of those three functions plotted: problem_graph

I'd like to know what's the value of the area, what are the integrals that reach that value and most importantly, since I cannot see it properly, which is the area that should be calculated between these three curves? (Below the red curve and above the blue curve? Below the yellow curve and above the blue curve? Below the yellow curve and above the red curve?)

Thanks for you attention!

Answer 1

I think it obvious that it can only be the area that is bounded by all three curves. If we denote

$$ y_1=8-6x+x^2\\ y_2=-4+2x\\ y_3=2+x $$

then we can express the area as

$$A=\int_1^2 (y_2-y_1)~dx + \int_2^6 (y_2-y_3)~dx$$

I'll assume you can handle it from here. FYI, I solved this as shown as well numerically by a completely different method and found that the area is $A=10\frac{1}{6}$.