Volume of a "tent"

Question

In $(x,y,z)$-space, the ground is the $(x,y)$-plane $z=0$. Above the ground is constructed a giant tent whose height over $(x,y)$ is $$ h(x,y)=z=\frac{100}{1+(x^2+4y^2)^2} $$ Find the volume enclosed by the tent (and the ground).

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To me, the $x^2+4y^2$ suggests a change of coordinate. However, I don't know how to approach problems like this. This is a previous exam question so things like Wolfram Alpha should be not used. Could anyone outline a chain of thoughts as how to proceed with this type of problems? Thanks.

Answer 1

$x^2+4y^2$ suggests indeed a change of coordinates. In particular, a shape: $$ z= \frac{100}{1+(x^2+y^2)^2} $$

would be circularly symmetric (so, in that case, what coordinates would you use?).

But in your exercise, instead of $y^2$, you have $4y^2$, that is, the $y$ are dilated of 1/2...so what is the ratio of the volumes of my shape and yours?

Answer 2

You can let $x = r\cos\theta$ and $y = 0.5r\sin\theta$. Next, realize that the surfaces bound will be your function $z=0$ and $z=f(x,y)$. The range of the bounds of the region $E$ will be from $0$ to $2\pi$ in $\theta$ and $0$ to $\infty$ in $r$ (think of it as a circle of "infinite radius"). Next, compute the Jacobian of the transformation and evaluate the iterated integral.