Here is a triple integral a student asked me, but which I can't seem to get right. I have tried to recalculate the integrals several times, but to no avail. If you're taking a Multivariable Calculus course at the moment, this may be a great exercise for you. Please help me to spot the mistake I simply can't spot.
Here is the original integral to be evaluated: Compute \begin{equation} \int_{-1}^1 \int_0^{x^2} \int_0^{1-y} dz\;dy\;dx. \end{equation} According to the limits of integration, the region whose volume is being sought is bounded by the $xy$-plane, the plane $z=1-y$, and the parabolic cylinder $y=x^2$. Evaluating the given integral in the given order yields $7/15$.
Switching the order between $dz$ and $dy$, I get \begin{equation} \int_{-1}^1 \int_0^{1-x^2} \int_{x^2}^{1-z} dy\;dz\;dx. \end{equation} I compute this integral as $8/15$, which is the given answer according to the student.
The original integral calculates to a value different from the given answer, but the (supposedly) equivalent integral evaluates to the correct answer. I used Wolfram Alpha to check against my hand-calculations and my calculations were right. So, I am suspecting that the limits of my integration in the second, equivalent integral may be problematic (but I have gone over it again and again) - yet the integral evaluates to the given answer!
I am curious as to where the flaw is...