$f(x)=|2-x|$
$g(x)=2-|2-x|$
Find the area of the region bounded by $f(x)$ and $g(x)$.
I was told that you actually need two definite integrals to solve this problem, since it involved absolute value, and $$\int_1^3 [g(x)-f(x)] dx$$ wasn't going to cut it. But I'm not exactly sure why. Instead, I would have to do: $$\int_1^2 [g(x)-f(x)] dx + \int_2^3 [g(x)-f(x)] dx$$ …or (since a cursory graph sketch reveals symmetry about the line $x=2$), $$2\int_1^2 [g(x)-f(x)] dx$$.
How do you go about solving these absolute value area-between-curves problems? They sound really tricky due to the presence of potential signage errors, and I would like to know how to solve them.
Best Answer
You can solve such questions easily by drawing the graph. After drawing the graph,you can find out the area of the enclosed region, which in this case is a square whose side length is $\sqrt{2}$ units,therefore the area is 2 sq. units.