find the volume bounded by the parabolic cylinder z=4-x^2 and the planes x=0, y=0, y=6 and z=0

Question

I just want some help figuring out the limits of this question :

So for z, we have to integrate from z=0 to z=4-x^2 (am i right?)

W.r.t. y, we should integrate from y=0 y=6 (am i right?)

w.r.t. x , can you please explain?

Thanks!

Answer 1

The volume we want is in the 1st octant, since all the coordinate planes form part of the boundary. The cylinder is sliced off at y=6
So the volume is just 6 times the area under the curve z = 4-x^2
v = 6∫[0,2] 4-x^2 dx
If you insist on doing a triple integral, then you are correct as far as you go. The last step is just to integrate x from 0 to 2, since we want z=0 on the parabola.