Form the double integrals of the case given

Question

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.

This solid is called a bicylinder. Some research into that term might help you visualize the solid and develop an integral. — Matthew Leingang, Jan 10 '18 at 23:48
Closely related to https://math.stackexchange.com/q/2599749/265466. — amd, Jan 11 '18 at 00:53
Aside from the existing links, this comment is to link this post as one of the (abstract) duplicates to the current choice of mother/target post. — Lee David Chung Lin, Jan 22 '19 at 10:28

score 0 · Answer 1 · answered Jan 10 '18 at 23:56

Here is a picture of your solid.

It is called a "Steinmetz solid"

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.

Form the double integrals of the case given

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