# Finding a volume in space

Hello, I need to find the volume between the plane z=1 and z=10-x*x-4*y*y.

I have tried using polar coordinates:
first integrate from z=1 to z=10-x*x-4*y*y
The I integrate in the plane z=1, but here I need to integrate over an ellipse, how do I do this? When I wrote the expression to polar coordinates I got an expression for r that I couldnt integrate.

I have also tried integrating over the ellipse using cartesian coordinates, but I got stuck aswell, can someone pleas help?

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Cyosis
Homework Helper
Could you show the actual function you have to integrate over? I suspect it has the form $\sqrt{9-4y^2}$? If so use the substitution $u=\frac{3}{2} \sin \theta$. Still integrating like this isn't very pretty. A more elegant method and by far the easiest to integrate is to notice that this is a paraboloid with elliptical cross section. You can then slice the paraboloid parallel to the x-y plane in a lot of slices and then integrate from z=1 to 10. The only thing you have to do is find an expression for the semi major axis and the semi minor axis as a function of z and use that the area of an ellipse is given by pi a b, with a the semi major axis and b the semi minor axis.

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HallsofIvy
Homework Helper
Could you show the actual function you have to integrate over? I suspect it has the form $\sqrt{9-4y^2}$? If so use the substitution $u=\frac{3}{2} \sin \theta$. Still integrating like this isn't very pretty. A more elegant method and by far the easiest to integrate is to notice that this is a paraboloid with elliptical cross section. You can then slice the paraboloid parallel to the x-y plane in a lot of slices and then integrate from z=1 to 10. The only thing you have to do is find an expression for the semi major axis and the semi minor axis as a function of z and use that the area of an ellipse is given by pi a b, with a the semi major axis and b the semi minor axis.
No, finding the volume bounded by z= 1 and $z= 10- x^2- 4y^2$ involves integrating the difference in z values, $(10- x^2- 4y^2)- 1= 9- x^2- y^2[/tex] over the circle [itex]x^2+ 4y^2= 9$, where the two surfaces cross.

Now the limits of integration, in xy-coordinates will involve something like $y= \frac{1}{2}\sqrt{9- x^2}$.

Hyper, you might consider using parameters r and $\theta$ such that $x= r cos(\theta)$ and $y= 4r sin(\theta)$, not standard polar coordinates, with r going from 0 to 1 and $\theta$ from 0 to $2\pi$. Do you know how to find the "differential of area" in r and $\theta$ using the Jacobian?

Cyosis
Homework Helper
Why wouldn't slices be applicable, it yields the same answer?