Negative area below x-axis and above $f(x)$

Question

I was finding the area below the x-axis and above $y = x^2 - 4x$. My outcome was a negative number (fair enough, it's under the x-axis), but wolframalpha for instance gives me a positive number (I get the same number but negative) which one is correct?

Answer 1

An area is always positive, occasionally an integral is negative. It depends on whether you want the area or the value of the integral.

Answer 2

Area is always positive, so you probably have counted integral $\int^b_a (y(x) - 0)dx$ but have to count $\int^b_a (0 - y(x))dx$ instead. To decide which sign to use for a function while counting area, you have to reason as follows.
If you have two functions $f(x)$ and $g(x)$ and need to count area between $f(x)$ and $g(x)$ from $a$ to $b$ $(a,b \in \mathbb R)$, then you may draw an approximate graph and see which of inequations points $(x,y)$ your area match: $y \leq f(x)$ means that $f(x)$ should be with sign $+$, as $f(x)$ is a minuend here. Similarly, $y \geq f(x) \Rightarrow f(x)$ should be with $-$, as $f(x)$ is a subtrahend here. $g(x)$ would have an opposite sign.
Consider that the functions may intersect repeatedly on $[a,b]$, in such case you have to define the area unambiguously.