Can the volume be non positive?

Question

I have a question please I have asked my teacher if the area could be negative she said :no never But what about the volume ? E.x If $y=x+2$ and $y=-x-2$,$x=0$ about x-axis The volume become zero!And about y-axis it becomes in negative ! Why ? Thanks all

Answer 1

Yes, volumes can be 0, but volumes can never be negative. The volume of a square is 0, for instance. You might want to look into measure theory and lebesgue measures.

Answer 2

It is the same as with money: An amount of money is never negative, but there are situations where some of the amounts occurring should be counted negative.

Now in geometric situations and the associated calculi it often occurs that the calculus counts certain areas, volumes, etc., automatically negative, and the user has to be aware of this. E.g., the volume of the prism $P$ spanned by three vectors ${\bf a}$, ${\bf b}$,${\bf c}\in{\mathbb R}^3$ can be determined by means of a certain determinant. This determinant sometimes turns out negative, even though the volume of $P$ is positive. When studying this phenomenon one finds out that the determinant carries some extra information, namely the orientation of the three given vectors with respect to the orientation of the standard basis vectors ${\bf e}_1$, ${\bf e}_2$, ${\bf e}_2$. Another example is the following: You want to find there area between the $x$-axis and the parabola $y=x^2-4$ for $-2\leq x\leq 2$. This area is positve, but the integral $\int_{-2}^2(x^2-4)\ dx$ set up to compute this area is negative.