Help with double integral - volume between 2 surfaces

[coodgee]

Homework Statement

find volume of the solid bounded by the surfaces

z = 1- [itex]\sqrt{\frac{x}{4}^2 + \frac{y}{2 sqrt{2}}^2}[/itex]

and x^2/4 -x +(Y^2)/2 = 0

and the planes z = 0 and z = 1

Homework Equations

z = 1- [itex]\sqrt{\frac{x}{4}^2 + \frac{y}{2 sqrt{2}}^2}[/itex]

and x^2/4 -x +(Y^2)/2 = 0

The Attempt at a Solution

I think the first surface is an ellipsoid with it's highest point at z =1 and x = 0 and y = 0 and the second "surface" I have interpreted as a cylinder whose base is an ellipse centred at x = 2 and y =0. So it seems like there could be two solids here, the first would have an elliptical base of the z = 0 plane and the top would be the surface of the first equation above and the sides would be the sides of the cylinder. But it seems like I could also have another similar solid where the top is the z =1 plane and the base is the surface of the first equation.

I think I must be a long way off track.

 

[haruspex]

I think the first surface is an ellipsoid

Not an ellipsoid. A cone, of sorts?
I agree the z=1 plane seems redundant.

 

[coodgee]

ok so we have now got it to the point where we are integrating from z goes from the xy plane up to the cone surface, and the area of integration is the elliptical cylinder.

so in polar coordinates I think the integrand is 1 -r/2

but I am not sure what the limits of integration are for r. The equation of the elliptical cylinder is ((x-2)/2)^2 + y^2/2 =1 so the ellipse doesn't have it's centre on the origin. does this mean the limits of integration for r is not 0 to 1?

 

[haruspex]

ok so we have now got it to the point where we are integrating from z goes from the xy plane up to the cone surface, and the area of integration is the elliptical cylinder.

I think it's a bit clearer if you take a slice at some z value. What does the region of integration look like?

 

[coodgee]

since the base we are integrating over is an ellipse with a = 2 and b = root2, we have converted to elliptic coordinates x = 2r cos(θ) and y = sqrt(2)r cos(θ)

and integrating the top function from my original post. when I convert it to elliptic coordinates it reduces to
(1 - r/2) r dr dθ

but now I'm not sure what the limits of integration for r should be.

I thought maybe I need to transform both funtions so the volume I am finding is centred at the origin?

Or is there a better way?

 

[haruspex]

since the base we are integrating over is an ellipse with a = 2 and b = root2, we have converted to elliptic coordinates x = 2r cos(θ) and y = sqrt(2)r cos(θ)

But what does the whole region look like at height z? The elliptical cylinder only provides one part of the boundary.