What are the bounds for evaluating a triple integral in the first octant?

[ProPatto16]

Homework Statement

evaluate triple integral of z.dV where the solid E is bounded by the cylinder y²+z²=9 and the planes x=0 and y=3x and z=0 in the first octant

Homework Equations

for cylindrical polar co-ords, x=rcos$$\theta$$, y=rsin$$\theta$$ and z=z

The Attempt at a Solution

im just struggling to grasp the bounds here. the cylinder has x as its centre line. and r=3. which means shape extends out from x 3 units along y and z axis's. and extends along x from origin 3x units. then stops due to plane on y. that's about as much as i can gather. the projected region that i should take the volume of the solid over should be projected onto the yz plane for this case. which would show a quarter circle with r = 3 right? with y=sqrt(9-z²) with y>0 so achieve first quadrant.
but i can't actually work out what to integrate each integral between.

help? much appreciated!

 

[HallsofIvy]

The given cylinder, $y^2+ z^2= 1$ has axis on the x-axis and its curved side projects to a circle in the yz plane. I would d you "swap" x and z in setting up the cylindrical coordinates:
$x= x$
$y= r sin(\theta)$
$z= r cos(\theta)$

Now the limits of integration should be easy.

 

[ProPatto16]

Yeah I knew that bit. R=3 and theta=pi/2 for first octant also. So looking at quarter circle is first quadrant of yz plane. Y goes from 0-3 and z goes from 0-3 and therefore with x=y/3 then x goes from 0 to 1?

 

[ProPatto16]

so then the integral needing evaulation is.. i can't do latex so ub means upper bounds and lb means lower bound

$$\int$$(ub 3, lb 0)$$\int$$(ub 3, lb 0)$$\int$$(ub 1, lb 0) z.dz.dy.dx??