Why is the area between these two functions positive?
\begin{align*}
f(x) &= (x-1)^3\\
g(x) &= (x-1).
\end{align*}
The area between $f(x)$ and $g(x)$ is in two parts.
The first part is in the 4th quadrant with lower limit $0$ and upper limit $1$, $f(x)$ is the upper function.
The second part is in the 1st quadrant with lower limit 1 and upper limit 2, $g(x)$ is the upper function.
When I compute the areas individually I get two equal positive numbers the sum of which is 0.5
Shouldn't the area in the fourth quadrant be negative? Or am I missing something?
I've been trying to understand why area in the fourth quadrant is positive.
Thank you.
Best Answer
Here is a graph of the two functions with f in pink and g in blue.
By symmetry, the area of each of the two enclosed regions is the same—area, here, meaning the geometric concept which is nonnegative (or perhaps positive), as opposed to signed area. In terms of signed area, one might think of the two pieces as having opposite sign because f is above g for oen piece and g is above f for the other piece. However, typically, a problem asking for the area between curves or the area enclosed by one or more curves is asking for geometric area. The typical method of solution in that instance is to consider each piece separately, integrating (top function) – (bottom function) for each piece, to guarantee a positive (nonnegative) result.