Volume of a Circle - Finding r with Double Integrals

In summary: To answer your first question, the integral has be transformed from Cartesian to polar coordinates. Rather than specifying the position of a point in terms of it's (x,y) coordinates, polar coordinates uses (r,Θ), where r is the distance from the origin to the point and Θ is the angle between the radius and the positive x semi-axis. For more information and answers to your subsequent questions see http://mathworld.wolfram.com/PolarCoordinates.html" .What makes you think its a cylinder?This is the full solution:http://users.on.net/~rohanlal/circ3.jpg http
  • #1
Ry122
565
2
http://users.on.net/~rohanlal/circle2.jpg [Broken]
this is part of the solution to finding the volume of a circle with double integrals.
I just want to know where the r from rdrd0 came from and also
why the limits on the d0 integral are 2pi and 0.
 
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  • #2
Ry122 said:
http://users.on.net/~rohanlal/circle2.jpg [Broken]
this is part of the solution to finding the volume of a circle with double integrals.
I just want to know where the r from rdrd0 came from and also
why the limits on the d0 integral are 2pi and 0.
I assume you mean this is part of a question to find the volume of the cylinder created by extruding a circle along the z-axis.

To answer your first question, the integral has be transformed from Cartesian to polar coordinates. Rather than specifying the position of a point in terms of it's (x,y) coordinates, polar coordinates uses (r,Θ), where r is the distance from the origin to the point and Θ is the angle between the radius and the positive x semi-axis. For more information and answers to your subsequent questions see http://mathworld.wolfram.com/PolarCoordinates.html" [Broken].
 
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  • #3
What makes you think its a cylinder?
This is the full solution:
http://users.on.net/~rohanlal/circ3.jpg [Broken]
 
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  • #4
Ry122 said:
http://users.on.net/~rohanlal/circle2.jpg [Broken]
this is part of the solution to finding the volume of a circle with double integrals.
I just want to know where the r from rdrd0 came from and also
why the limits on the d0 integral are 2pi and 0.
This is the volume of a sphere, not a circle- circles don't have "volume"!

And you should have learned that the "differential of area in polar coordinates" is [itex]r dr d\theta[/itex] when you learned about integrating in polar coordinates. There are a number of different ways of showing that. I recommend you check your calculus book for the one you were expected to learn.
 
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  • #5
and why is the limit 2pi to 0?
 

1. How is the volume of a circle calculated using double integrals?

The volume of a circle can be calculated using a double integral by integrating the function r^2 over the circle's region. This means that the integral is taken over the entire circle, with r as the variable. The result of the integral is then multiplied by 2π, the circumference of a circle, to get the total volume.

2. What is the difference between a single integral and a double integral when finding the volume of a circle?

A single integral is used to find the area under a curve, while a double integral is used to find the volume under a surface. In the case of finding the volume of a circle, a double integral is necessary because the circle is a two-dimensional shape.

3. How does the formula for the volume of a circle with a double integral relate to the formula for the volume of a sphere?

The formula for the volume of a circle with a double integral is a special case of the formula for the volume of a sphere. Both formulas involve the variable r, which represents the radius of the circle or sphere. However, the formula for the volume of a sphere also includes the variable z, which represents the height of the sphere.

4. Can double integrals be used to find the volume of other shapes besides circles?

Yes, double integrals can be used to find the volume of any three-dimensional shape. The key is to determine the appropriate function to integrate over the region of the shape, which may vary depending on the shape's dimensions and geometry.

5. Is finding the volume of a circle with a double integral a more accurate method compared to other methods?

The method of using a double integral to find the volume of a circle is not necessarily more accurate than other methods, but it can be a useful approach for certain situations. Other methods, such as using the formula for the volume of a cylinder, may be more efficient in some cases. It ultimately depends on the specific problem at hand and the individual's understanding and comfort with various mathematical techniques.

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