Find volume of this object using integrals

In summary, the conversation is about using multiple integrals to determine the volume of a 3D object. The person has tried using cartesian, cylindrical, and spherical coordinates but has difficulty integrating. They are seeking help with writing the limits of integration correctly. The final result is supposed to be 32/9. They also discuss the placement of origin in cylindrical coordinates and the equation for a circle in polar coordinates.
  • #1
ananonanunes
18
6
Homework Statement
_
Relevant Equations
_
1685030223409.png


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)

1685030427777.png


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)
 
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  • #2
ananonanunes said:
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...
 
Last edited:
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  • #3
ananonanunes said:
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?
 
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  • #4
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 [itex]x^2 + (y-1)^2 = 1[/itex].

The first of these leads to a simple integral; the other does not.
 
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  • #5
SammyS said:
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
 
  • #6
vela said:
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 ; 0θ≤π
 
  • #7
pasmith said:
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 [itex]x^2 + (y-1)^2 = 1[/itex].

The first of these leads to a simple integral; the other does not.
I understand what you mean, thanks for the help
 

Related to Find volume of this object using integrals

How do you set up an integral to find the volume of a solid of revolution?

To set up an integral for finding the volume of a solid of revolution, you typically use the disk or washer method for rotation around the x-axis or y-axis. For the disk method, if the function is y = f(x) and the solid is rotated around the x-axis, the volume V is given by the integral \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \). For the washer method, if there are two functions y = f(x) and y = g(x), the volume is \( V = \pi \int_{a}^{b} \left( [f(x)]^2 - [g(x)]^2 \right) \, dx \).

What is the difference between the disk method and the shell method?

The disk method involves slicing the solid perpendicular to the axis of rotation and summing up the volumes of the disks or washers formed. The shell method involves slicing the solid parallel to the axis of rotation and summing up the volumes of cylindrical shells. For the shell method, if the function is y = f(x) and the solid is rotated around the y-axis, the volume V is given by \( V = 2\pi \int_{a}^{b} x f(x) \, dx \).

How do you find the volume of a solid bounded by two curves?

To find the volume of a solid bounded by two curves using the washer method, you first identify the outer radius R(x) and the inner radius r(x) of the washers formed by rotating the curves around an axis. The volume V is then given by the integral \( V = \pi \int_{a}^{b} \left( [R(x)]^2 - [r(x)]^2 \right) \, dx \). The limits of integration a and b are the points where the two curves intersect.

Can you use integrals to find the volume of solids with non-circular cross-sections?

Yes, integrals can be used to find the volume of solids with non-circular cross-sections by integrating the area of the cross-sections along the axis of the solid. If A(x) represents the area of the cross-section at position x, the volume V is given by \( V = \int_{a}^{b} A(x) \, dx \). This method is often used for solids with known cross-sectional shapes like rectangles or triangles.

What are common mistakes to avoid when finding volumes using integrals?

Common mistakes include: 1. Incorrectly setting up the limits of integration.2. Misidentifying the radius or height functions in the disk,

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