Sorry for the vagueness here, as I'm trying to remember a problem I worked on a week ago. The problem involved evaluating a triple integral in spherical coordinates, $\displaystyle \iiint\limits_V (equation) d\rho d\phi d \theta$, where $V$ is the region bounded by a sphere of radius $1$ centered at the origin and the cone $\phi = 5\pi/6$. When I first evaluated the integral I used $0 \le \phi \le 5\pi/6$ as my bounds for $\phi$ and got a negative answer, and so I thought that I'd made a mistake and changed my bounds to $5\pi/6 \le \phi \le \pi$, which gave me the absolute value of my first answer. However, the teacher says that the answer (which I got with my first bounds for $\phi$) is indeed negative, so is this in fact possible?
[Math] Is it possible for a triple integral to give a negative result
calculus
Best Answer
For the sake of giving you some intuition, let $V=[0,1\textbf{]}^3$ and consider $$\iiint \limits_V-1\,dxdydz$$
The answer: yes, it is possible.