Triple Integration of a Strange Cylinder

In summary, the conversation discusses finding the volume of a region defined by a mathematical equation. The individual seeking help has tried using cylindrical polars and the Jacobian, but has encountered an issue with their integration. Another person provides a hint, explaining that the volume is symmetric about the x-axis and that the true volume is twice the volume above the z-axis. The individual expresses gratitude for the hint and apologizes for not using Latex code.
  • #1
Rocker222
2
0
Hey all,

If anyone has some hints on how to do this one it would be much appreciated:

Find the volume of the region given by x2 + y2 ≤ a2, 0 ≤ z ≤ x.

So I've gone to cylindrical polars, and threw in the Jacobian, r. If I integrate with my bounds being:

0 to a, 0 to pi, 0 to rcos(theta) of r dz d(theta) dr I'm getting 0 which can't be. What is it that I'm missing here. Any hint would be great.
 
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  • #2
So you are integrating
[tex]\int_{r=0}^a\int_{\theta= 0}^\pi\int_{z= 0}^{r cos(\theta)} r dzd\theta dr[/itex]?

If you look closely at this situation you will see that this is symmetric about the x-axis and so, with z= x as a boundary, half the volume is below the z= 0 plane, half above. Since your integral treats the volume below the z= 0 plane as negative, those cancel. The true "volume" is twice the volume above the z-axis. You get the volume above the z= 0 plane by assuring that x is positive: by taking [itex]\theta[/itex] from [itex]-\pi/2[/itex] to [itex]\pi/2[/itex].
 
  • #3
Thank you very much. That's just the bit I needed. Sorry about not using the Latex code. I'm new to this site.
 

Related to Triple Integration of a Strange Cylinder

1. What is a Strange Cylinder?

A Strange Cylinder is a three-dimensional shape that has a circular base and a curved surface that is not like a traditional cylinder. It may have a varying radius or curvature, making it a unique shape to integrate.

2. Why is triple integration used for a Strange Cylinder?

Triple integration is used to calculate the volume of a three-dimensional shape, such as a Strange Cylinder. Since a Strange Cylinder has a curved surface, it cannot be easily defined by simple geometric formulas, making triple integration the most accurate method for finding its volume.

3. What are the steps to triple integrate a Strange Cylinder?

The first step is to identify the limits of integration for each variable (x, y, and z). Then, set up the triple integral using the appropriate formula for the shape of the Strange Cylinder. Next, evaluate the integral by plugging in the limits of integration and solving for the volume. Finally, double check your work and make any necessary adjustments.

4. Can triple integration be used for any type of Strange Cylinder?

Yes, triple integration can be used for any type of Strange Cylinder, as long as the limits of integration are properly identified and the appropriate formula is used. This method is flexible and can be applied to various shapes and sizes of Strange Cylinders.

5. Are there any real-world applications for triple integration of a Strange Cylinder?

Yes, triple integration of a Strange Cylinder has various real-world applications, such as calculating the volume of industrial tanks, chemical vats, and even certain types of food containers. It is also used in fields such as engineering, physics, and architecture to accurately measure and design three-dimensional objects with curved surfaces.

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