Finding Volume of Solid in First Octant with Bounded Polar Equations

[andromeda92]

Homework Statement

find the volume of the solid bounded by the graphs of the given equations:

r=1+cos∅
z=y
z=0
(the first octant)

Homework Equations

V=$\int$$\int$ rdrd∅

The Attempt at a Solution

So, I've been having trouble deciding what to integrate from and to. I converted the z equations into polar coordinates, so, z=rsin and z=0

$\int$$^{2\pi}_{0}$$\int$$^{1+cos∅}_{0}$ (rsin)rdrd∅

Mod note: revised LaTeX
$$\int_0^{2\pi} \int_0^{1 + cos(\theta)} rsin(\theta) r~dr~d\theta$$

I found, from the first integration, [2∏]\int[/0]$r^(3)sin∅/3 d∅ and I need to plug in (1+cos∅) and 0 for r...but, this seems WAY too complicated having a trig function to the 3rd power.<br /> <br /> I'm not even sure if I have the beginning correct. Can anyone help me out?$

 

[Astronuc]

In three dimensions, the first quadrant has x ≥ 0, y ≥ 0, z ≥ 0, and think about the limits of ∅ for x ≥ 0, y ≥ 0.

Think about this - http://www.wolframalpha.com/input/?i=polar+plot+r=1+cos+theta (don't worry about the angle being theta)

And do you want to use cylindrical or spherical coordinates?