Triple Integration Help

Pi*sqrt(3) In summary, the conversation is about finding the volume of a solid B that is bounded by two surfaces and lies inside the surface x^2 + y^2 = 1. The attempt at a solution involves using the projection onto the xy and zy planes and converting the integral into cylindrical coordinates. The final solution is Pi*sqrt(3) but it does not match the answer using the formula for the volume of a cylinder.
  • #1
Chibus
5
0

Homework Statement


Sketch the solid B that lies inside the surface x^2 + y^2 = 1 and is bounded above and below by the surface x^2 + y^2 + z^2= 2^2. Then find the volume of B.


Homework Equations



projxy = projection onto the xy plane, proj zy = projection on the zy plane

The Attempt at a Solution


(See attached)

http://img511.imageshack.us/img511/440/chibusq.jpg

I just wanted to check whether my definition of the integration is correct, meaning:

1) Is the function of the integration right? (Since the circle on the xy plane is x^2 + y^2 = 1, I've used that)

2) Are the limits correct?

Thanks for any help!
 
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  • #2
Any help?

I've converted the above integral into cylindrical coordinates and solved it, and I ended up with Pi*sqrt(3) as the solution. However, it doesn't match the answer of solving the cylinder by using the formula V(cyl)=pi*r^2*h = pi*(1)^2*2*sqrt(3) = pi*2*sqrt(3)
 

1. What is triple integration and why is it important in science?

Triple integration is a mathematical technique used to find the volume of a three-dimensional region. It is important in science because many physical phenomena, such as fluid flow and electromagnetic fields, are described in three dimensions and require triple integration for accurate analysis and predictions.

2. How do I set up a triple integral?

The setup of a triple integral depends on the type of region being integrated over. Generally, the triple integral is written as ∫∫∫f(x,y,z)dV, where f(x,y,z) is the function being integrated and dV represents the volume element. The limits of integration for each variable must also be specified, which can be determined based on the boundaries of the region.

3. What are some common methods for solving triple integrals?

Some common methods for solving triple integrals include using Cartesian, cylindrical, and spherical coordinates. These coordinate systems allow for simpler expressions and limits of integration, making the integration process easier. Another method is to break up the region into smaller, simpler subregions and integrate over each separately.

4. How can I check if my triple integral is correct?

One way to check if a triple integral is correct is to use the divergence theorem, which states that the triple integral of a vector field over a region is equal to the flux of the field through the boundary of the region. If the calculated value matches the flux, then the triple integral is likely correct. Another method is to use software or a graphing calculator to visualize the three-dimensional region and compare it to the calculated volume.

5. Can triple integration be used in other fields besides science?

Yes, triple integration can be used in fields such as engineering, economics, and finance. In engineering, it can be used to find the mass, center of mass, and moment of inertia of three-dimensional objects. In economics and finance, it can be used to calculate the value of three-dimensional investments or to analyze three-dimensional data sets.

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