The volume is actually
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 , and the parabolic cylinder .
However, there's something incorrect here since the correct answer is 16.
Can anyone see a mistake anywhere? (especially with my limits!).
There's one more thing I don't quite understand.
I evaluated the integrand w.r.t z on the left and you do indeed get what's written on the right. However, geometrically, what does it mean when we integrate 1 w.r.t z?
To me it seems to suggest that it's the volume of a unit cube, but the whole point of triple integrals is that the volume of these cubes is made really small (ie. side length ). If that is the case, I would expect to be evaluating a function of a variable w.r.t z and not just 1.
In Cartesian coordinates the elementary volume unit is dx dy dz.
To get the volume you need to integrate dx dy dz over the boundaries.
Lets's go to physics. When a solid has a non-uniform density you can evaluate its mass through
When the density is uniform then the mass is where V is the volume of the solid.
I hope this answers your question