Find volume of this object using integrals

[ananonanunes]

Homework Statement

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Relevant Equations

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I am given this expression which represents an object in 3D and the goal is to determine its volume using multiple integrals.
I started by drawing what I think is the object as well as two "slices" of that object on different planes (z=2 and z=1)

I have tried using cartesian, cylindrical and spherical coorddinates but I get a final result that seems impossible to integrate. My first instinct was to try to use Cavalieri's principle but I can't find a way to write the area of each "slice" only in terms of z. I think the problem is I'm not writing the limits of integration correctly so if anyone could help me out with that, it would be much appreciated.

(Final result is supposed to be 32/9)

 

[SammyS]

Homework Statement: _
Relevant Equations: _

View attachment 327066

I am given this expression which represents an object in 3D and the goal is to determine its volume using multiple integrals.
I started by drawing what I think is the object as well as two "slices" of that object on different planes (z=2 and z=1)

View attachment 327067

I have tried using cartesian, cylindrical and spherical coordinates but I get a final result that seems impossible to integrate. My first instinct was to try to use Cavalieri's principle but I can't find a way to write the area of each "slice" only in terms of z. I think the problem is I'm not writing the limits of integration correctly so if anyone could help me out with that, it would be much appreciated.

(Final result is supposed to be 32/9)

It looks to me like cylindrical co-coordinates will work.

In what order did you do the integration, when you tried them?

Added in Edit:
Maybe not...

 

[vela]

I think the problem is I'm not writing the limits of integration correctly so if anyone could help me out with that, it would be much appreciated.

Can you write down the equation for the circle in polar coordinates?

 

[pasmith]

There are two options:
1) You can place the origin of the cylindrical coordinates at the origin.
(2) You can place the origin of the cylindrical coordinates at the centre of the circle x^2 + (y-1)^2 = 1.

The first of these leads to a simple integral; the other does not.

 

[ananonanunes]

It looks to me like cylindrical co-coordinates will work.

In what order did you do the integration, when you tried them?

Added in Edit:
Maybe not...

I tried integrating first z then r then teta because i wrote z in terms of r and r in terms of teta

 

[ananonanunes]

Can you write down the equation for the circle in polar coordinates?

I think that might be where my problem was but now that I look at it again it should be something like 0≤r≤2sinθ ; 0≤θ≤π

 

[ananonanunes]

There are two options:
1) You can place the origin of the cylindrical coordinates at the origin.
(2) You can place the origin of the cylindrical coordinates at the centre of the circle x^2 + (y-1)^2 = 1.

The first of these leads to a simple integral; the other does not.

I understand what you mean, thanks for the help