Finding the area between two curves

[Tyler90]

Hello,

When finding the area between two curves how do I know which curve to use as f(x) and which as g(x), or how to determine which is 'larger' over the interval.
Particularly for the graphs f(x)=(x-1)(x-2) and f(x)=(3(x-1))/x

[fun_jirachi]

Hey there!

Usually a rough sketch will suffice - in this case, we have that \(\frac{3(x-1)}{3}\) being a hyperbola such that \(y \neq 3, x \neq 0\) (ie. asymptotes \(y=3, x=0\)). We can also see that \((x-1)(x-2)\) is a parabola with intercepts \(x=1, 2\). This sketch should indicate roughly that the area between the two curves is bounded by two points of intersection to the right of the x-axis, and that \(\frac{3(x-1)}{3}\) is larger over the bounded interval. You can then algebraically find the upper and lower bounds for this area.

Alternatively, you could consider using test points between the two points of intersection you've found - over some domain \([a, b]\) between two curves \(f(x), \ g(x)\), if the area is bounded and if \(\exists x \in (a, b) \text{ such that } f(x) > g(x), f(x) \geq g(x) \ (\forall x \in [a, b])\). Essentially, we can just use a test point to find which curve is bigger over the upper and lower bounds, by subbing into both equations, and whichever is larger will remain larger over the domain bounded by the upper and lower bound you've found originally.

Hope this makes sense