Calculus 3 Multi-Integration (volume)

[IAmGroot48]

Homework Statement

In this particular question the following is stated.

Find the volume of the region below z=4(x²+y²), above z=0, and between the two cylinders x²+y²=1² and x²+y²=3².

Relevant Equations

This question doesn't necessarily say if the two cylinders are intersecting each other, so I'm not 100% clear how to go on about this question. If I could get an Idea on how to set this up or solve it. I'd be able to figure out the rest.

Pending

 

[PeroK]

I'm not sure where the ambiguity is. The cylinders are concentric and the volume in question lies between them.

 

[IAmGroot48]

I feel like I may be over thinking this--I have ADHD so sometimes I have to read things over and over again to imagine what I am trying to do. Am I suppose to use triple integration for this or is this something double integration can do. What throws me off is the cylinder within the cylinder and then the region below z=4(x²+y²). I can't grasp where I am suppose to begin.

 

[berkeman]

I feel like I may be over thinking this--I have ADHD so sometimes I have to read things over and over again to imagine what I am trying to do. Am I suppose to use triple integration for this or is this something double integration can do. What throws me off is the cylinder within the cylinder and then the region below z=4(x²+y²). I can't grasp where I am suppose to begin.

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)...

EDIT -- Mark mentions a double integral below, so I could be wrong about the simplification to a single integration...

 

[Mark44]

I feel like I may be over thinking this--I have ADHD so sometimes I have to read things over and over again to imagine what I am trying to do. Am I suppose to use triple integration for this or is this something double integration can do. What throws me off is the cylinder within the cylinder and then the region below z=4(x²+y²). I can't grasp where I am suppose to begin.

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.

The volume of the enclosed region can be calculated with a double integral.

 

[IAmGroot48]

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This is what I am getting as far as the sketch. The paraboloid does indeed touch the origin, and then the surface z=0 is shown as a plane at well z=0. so from z=0 to z=4(x²+y²) is the first integration? In other words does the set up look like 4(x²+y²) dydx? and the first integration is from x²+y²=1² (bottom) to x²+y²=3² (top)?

Just Fyi: I am not the type of person who just wants the answer. I want to be able to find the answer myself for the most part but I want to understand why or how.

 

[IAmGroot48]

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.

The volume of the enclosed region can be calculated with a double integral.

Was this rhetorical?

 

[PeroK]

Was this rhetorical?

Parabolic!

 

[IAmGroot48]

I apologize I'm not sure what you're meaning when you only say parabolic.

 

[PeroK]

I apologize I'm not sure what you're meaning when you only say parabolic.

Only that it's a parabolic question, rather than a rhetorical one.

 

[IAmGroot48]

I feel there was a funny thrown in there, but its almost got me thinking...slightly sparked an ember...

 

[PeroK]

I suggest you look at cylindrical coordinates and figure our the range for $r$ or $\rho$, whichever you use, for each value of $z$.

 

[berkeman]

This is what I am getting as far as the sketch.

Wow, nice sketch! What software package did you use to do that?

 

[IAmGroot48]

View attachment 293873|

This is what I am getting as far as the sketch. The paraboloid does indeed touch the origin, and then the surface z=0 is shown as a plane at well z=0. so from z=0 to z=4(x²+y²) is the first integration? In other words does the set up look like 4(x²+y²) dydx? and the first integration is from x²+y²=1² (bottom) to x²+y²=3² (top)?

Just Fyi: I am not the type of person who just wants the answer. I want to be able to find the answer myself for the most part but I want to understand why or how.

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.

 

[IAmGroot48]

I suggest you look at cylindrical coordinates and figure our the range for $r$ or $\rho$, whichever you use, for each value of $z$.

Ill show you the work I've done so far. I think it finally clicked. "I think"

 

[IAmGroot48]

Also...I am a firm believer every single website should have a Dark Mode from this point on 2022 and above. The bright white colors just murder my eyes...

-A Computer Engineer Major

 

[berkeman]

We used to have different skins that you could choose from. Let me do a little searching...

 

[Mark44]

Was this rhetorical?

No, not at all. I didn't know you had access to a graphing tool.

 

[berkeman]

.I am a firm believer every single website should have a Dark Mode from this point on 2022 and above.

I spoke with the Admin, and he will look at adding a Dark Mode in the 2022 updates.

 

[IAmGroot48]

Oh you rock! Thats awesome! I truly believe all websits should have that option. We're moving into a technological generation where a huge majority of our hardware is being ran by software so i feel we need to have dark mode to lessen the health concerns to our eyes. Sounds insane but a huge help to our optical system. Thank you.

 

[IAmGroot48]

As for this question. This is what I came up with.

 

[Mark44]

I am a firm believer every single website should have a Dark Mode from this point on 2022 and above. The bright white colors just murder my eyes.

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.

 

[IAmGroot48]

Yeah I am on the computer almost all day, so I need to take breaks from time to time. So even when I am learning to code I put my whole visual studio on a dark type format. I also use a software called Rainmeter to control all of my computers setting. Its an overlay for your whole desktop setting.

 

[PeroK]

As for this question. This is what I came up with.

Or, in Latex: $$V = \int_0^{2\pi}\int_1^3\int_0^{4\rho^2}\rho \ dz \ d\rho \ d\phi = 160 \pi$$
https://www.physicsforums.com/help/latexhelp/