I've drawn a nice picture (essential) and I here's what i've deduced:Find the volume of the solid in the first octant bounded by the coordinate planes, the plane $\displaystyle x=3$, and the parabolic cylinder $\displaystyle z=4-y^2$.

$\displaystyle \int_0^3 \int_0^x 4-y^2 \ dy \ dx$

$\displaystyle \int_0^3 \left[4y-\frac{y^3}{3}\right]_0^x $

$\displaystyle \int_0^3 4x-\frac{x^3}{3}dx$

$\displaystyle \left[2x^2-\frac{x^4}{12}\right]^3_0$

$\displaystyle 9\left(2-\frac{9}{12}\right)$

$\displaystyle =\frac{45}{4}$

However, there's something incorrect here since the correct answer is 16.

Can anyone see a mistake anywhere? (especially with my limits!).