Deducing the volume of an elliptical cone

[yaakob7]

Here again

Homework Statement

Find the volume of a right elliptical cone with an elliptic base with semi-axes a and b and heigh h

Homework Equations

So: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The Attempt at a Solution

That's what I have, but answer should be:

$V=\frac{1}{3}abh\pi$

I've checked it all over again like 10 times, but I can't find the mistake. If you can see it I'd be grateful

 

[LCKurtz]

To tell you the truth, I can 't bring myself to slog through all your steps. But I have a question for you. Have you had change of variables in double integrals yet? For example, do you know how to show the area of the ellipse $$\frac {x^2}{a^2}+\frac{y^2}{b^2}=1$$ is $\pi a b$ by mapping the ellipse to a circle? The reason for my asking is that if you have studied that, a much easier way to do the problem is to use the elliptical cross-sections. You can figure out the equation of the ellipse cross section at height $z$ for $0\le z \le h$ and either use the above formula or develop it with the appropriate change of variables.

 

[yaakob7]

To tell you the truth, I can 't bring myself to slog through all your steps. But I have a question for you. Have you had change of variables in double integrals yet? For example, do you know how to show the area of the ellipse $$\frac {x^2}{a^2}+\frac{y^2}{b^2}=1$$ is $\pi a b$ by mapping the ellipse to a circle? The reason for my asking is that if you have studied that, a much easier way to do the problem is to use the elliptical cross-sections. You can figure out the equation of the ellipse cross section at height $z$ for $0\le z \le h$ and either use the above formula or develop it with the appropriate change of variables.

I know how to make an implicit differentation but, I can't do that to an intregal expression.

 

[LCKurtz]

To tell you the truth, I can 't bring myself to slog through all your steps. But I have a question for you. Have you had change of variables in double integrals yet? For example, do you know how to show the area of the ellipse $$\frac {x^2}{a^2}+\frac{y^2}{b^2}=1$$ is $\pi a b$ by mapping the ellipse to a circle? The reason for my asking is that if you have studied that, a much easier way to do the problem is to use the elliptical cross-sections. You can figure out the equation of the ellipse cross section at height $z$ for $0\le z \le h$ and either use the above formula or develop it with the appropriate change of variables.

I know how to make an implicit differentation but, I can't do that to an intregal expression.

What I asked you has nothing to do with implicit differentiation. Just tell me this: Are you allowed to use the area formula $\pi ab$ for your standard xy ellipse area as a given? If the answer to that is yes, then figure out the equation of the elliptical cross section of your cone at height $z$ for $0\le z \le h$ and use that formula for its area. Then you can integrate the elliptical cross section area as a function of z to get the volume.