Initially I would think you could just do a single integration along z because of the symmetry of the surfaces. Can you try making a couple sketches to help you understand the geometry? Do one sketch from the side (say, looking down the x-axis with the y-axis to the right and z upward), and do one sketch in a perspective view looking at an angle down from the side (like along the line x=y=z)...
Can you sketch a graph of the surface ##z = 4(x^2 + y^2)##? It's a paraboloid with its vertex at the origin, its axis along the z-axis, and opening upward. The two cylinders also have their axes along the z-axis.
Was this rhetorical?
Parabolic!
Only that it's a parabolic question, rather than a rhetorical one.
Wow, nice sketch! What software package did you use to do that?
Normally I use autocad. But this is a simple piece of software you can easily get for free. You can get it on Microsoft Store and it's called Graphing Calculator 3, there's a free version and a pro version.
Ill show you the work I've done so far. I think it finally clicked. "I think"
No, not at all. I didn't know you had access to a graphing tool.
I spoke with the Admin, and he will look at adding a Dark Mode in the 2022 updates.
My computers have an option to turn down the brightness. Have you tried that?Personally, I really dislike dark mode, as I find it harder to read, but hey, different strokes for different folks.
Or, in Latex: $$V = \int_0^{2\pi}\int_1^3\int_0^{4\rho^2}\rho \ dz \ d\rho \ d\phi = 160 \pi$$
