TITLE: Intuitively, why does $\int_{-\infty}^{\infty}\sin(x)dx$ diverge?
QUESTION [8 upvotes]: According to Wolfram Alpha, $\int_{-\infty}^{\infty}\sin(x)dx$ does not converge.
This makes no sense to me, intuitively, which I'll justify with a plot:

As we see, the positive and negative areas 'cancel out', so, for any $\alpha \in \mathbb{R}$, $\int_{-\alpha}^{\alpha}\sin(x)dx=0$ (I'm just thinking geometrically- in no way is this supposed to be a rigourous justification).
So, why is is that $\int_{-\infty}^{\infty}\sin(x)dx$ diverges?
A natural and intuitive reason, more than a rigourous one, would be best.
Thanks

REPLY [3 votes]: The areas cancel out because you've chosen to draw it that way. You could have drawn it with a green upper lobe at each end in which case you'd say there's more green than red and they don't cancel. The value of the integral needs to be independent of the little piece you've chosen to draw.