Form the double integrals to calculate the volume of an object inside the cylinder $x^2+z^2=a^2$ and $x^2+y^2=a^2$

I have a trouble to form the double integrals of this problem. The first one I can't imagine what is the base plane of the object so I can't determine the lower and upper bound of each integrals. Also, now how to determine the function we need to integrate? I have seen some articles discussing about double integration problems, but I don't have any idea to solve this one. Do I need to form $z = \pm \sqrt{a^2 - x^2}$ or what? Please help. Regards.


Here is a picture of your solid.

It is called a "Steinmetz solid"

enter image description here

Convert to cylindrical. Since $x$ is in both equations let x be "odd man out."

$z = r\cos \theta\\ y = r\sin \theta\\ x = x\\ dx\ dy\ dz = r\ dx\ dr\ d\theta$

$x^2 = a^2 - r^2 \cos^2\theta$ when $\cos \theta \ge \sin\theta$ and $x^2 = a^2 - r^2 \sin^2\theta$ otherwise.

Use symmetry to your advantage

And that should be enough to get you on your way.


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