How to determine which function is the "higher" one of two from equations?

Question

I'm wondering how I can determine from the equations of two functions without graphing which function will be the "upper" function when calculating the area between the two. For example, the equations:

$$f(x) = x^2$$ $$g(x) = 1 - x^2$$

Graphing it is is obvious $g(x)$ is the "higher" of the two; meaning the area between the curves would be calculated as $\int_a^b(1-x^2) - \int_a^b x^2$. The graph is shown below:

Knowing this, how would one find mathematically which is the first function that appears in the calculation? Would absolute values make this distinction null, meaning an absolute value would mean the subtraction can occur regardless of order of functions? How would you rationalize using such an operation?

Thank you!

Answer 1

You can find the intersection points of the two curves and see which curve is the higher curve in each of the intervals determined by the intersection points. Alternatively, to calculate the area bounded by two curves, compute $\int |f(x)-g(x)| dx$, which ensures you are taking the positive area in each of these intervals. Note that this is not necessarily the same as $|\int (f(x)-g(x)) dx|$!

Answer 2

If you find the points $a<b$ where $f(x)=g(x)$ and the two function are continuous on $(a,b)$ than the you can find the sign of $f(c)-g(c)$ at any $c\in(a,b)$ and if this is $>0$ than $f(x)$ is ''higher'' than $g(x)$ in the interval. Otherwise if the sign is negative.