How to interpret a definite integral outside the domain of the function?

Question

Considering the geometric interpretation of the definite integral (finding the area under a curve) what should I do if the interval of integration is not contained in the function's domain? For instance

$$ \text{Calculate} \int_{-2}^4 f(x)dx \quad \text {where} \quad f(x)= \begin{cases} x \quad \text{if} \ 0\le x \le 2 \\ 2x \quad \text{if} \ 2 \le x \le 4 \end{cases} $$

Should I consider only the interval where the function is well defined?

If I just had to calculate it, I would say that the function is not integrable over that interval, but since the important thing in this case is the area interpretation I'm not shure what to do.

Answer 1

There is no sensible way to define $\int_a^b f(x)dx$ if $[a,b]$ is not inside the domain of $f$. My guess is that the question writer had intended to say $f(x) = x$ if $0 \leq x \leq 2$, and $f(x) = 2x$ if $2 < x \leq 4$, and $f(x) = 0$ otherwise.