Finding the Volume of a Region

[Big-oh]

Homework Statement

"Find the volume of the region bounded by (x²+y²+z²)² = x"

Homework Equations

--> Need to get proper bounds on ρ, ∅, θ and evaluate triple integral.

The Attempt at a Solution

I have tried converting this into the spherical and cylindrical co-ordinates, but have yet to get any bounds at all...

When using spherical co-ordinates, the equation turns into:

ρ³ = sin∅cosθ

which doesn't give bounds for anything..

and similarly, if I use the cylindrical coordinates:

(r²+z²)² = rcosθ

which doesn't give me bounds either..

Am I missing something here?

 

[Mark44]

Homework Statement

"Find the volume of the region bounded by (x²+y²+z²)² = x"

Homework Equations

--> Need to get proper bounds on ρ, ∅, θ and evaluate triple integral.

The Attempt at a Solution

I have tried converting this into the spherical and cylindrical co-ordinates, but have yet to get any bounds at all...

When using spherical co-ordinates, the equation turns into:

ρ³ = sin∅cosθ

which doesn't give bounds for anything..

Sure it does. -1 ≤ sin($\phi$) ≤ 1, and the same for cos(θ), so -1 ≤ ρ ≤ 1, although it might be that ρ ≥ 0 by convention.

and similarly, if I use the cylindrical coordinates:

(r²+z²)² = rcosθ

which doesn't give me bounds either..

Am I missing something here?

 

[Dick]

If you actually try to do it using Mark44's comment in spherical coordinates you are going to get an awfully nasty integral. Use cylindrical. And here's a hint. Rotate the cylindrical coordinate system so the axis lies along the x-axis instead of the z-axis. Or you can think of this as rotating the volume. Now you want the volume of (r^2+z^2)^2=z. Makes life much easier.