- #1
erzagildartz
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how to solve triple integrals in cylindrical, spherical and rectangular coordinates ..easy ways
How to Integrate Triple Integrals in Different Coordinate Systems?
- Thread starter erzagildartz
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In summary, integrating triple integrals in different coordinate systems involves converting the limits of integration and the integrand into the desired coordinate system. This can be done using transformation formulas and understanding the geometry of the coordinate system. It is important to consider the orientation of the coordinate system and to make sure the limits of integration are consistent with the chosen system. The integration process is similar to that of double integrals, but with an additional dimension. It is also possible to use triple integrals to calculate volume, mass, and other physical quantities in three-dimensional space.
- #2
Mr-R
- 123
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Welcome to PF
You should provide us with your mathematical background first, right?
What do you mean by easy ways?
Well, you just integrate each of the components normally one at a time. Depending on your limits.
Assuming you have Cartesian coordinates (x,y,z). Convert each component to its Spherical/Cylindrical counterpart and multiply by the Jacobian of the new coordinate system. So, just convert to the new coordinate system, multiply by the Jacobian and then integrate normally.
Check this link, it has some nice examples for you:
http://mathinsight.org/triple_integral_introduction (read this first)
http://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords.aspx
You should provide us with your mathematical background first, right?
What do you mean by easy ways?
Well, you just integrate each of the components normally one at a time. Depending on your limits.
Assuming you have Cartesian coordinates (x,y,z). Convert each component to its Spherical/Cylindrical counterpart and multiply by the Jacobian of the new coordinate system. So, just convert to the new coordinate system, multiply by the Jacobian and then integrate normally.
Check this link, it has some nice examples for you:
http://mathinsight.org/triple_integral_introduction (read this first)
http://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords.aspx
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FAQ: How to Integrate Triple Integrals in Different Coordinate Systems?
What are the steps to solving a triple integral?
The steps to solving a triple integral are as follows:
- Identify the limits of integration for each variable.
- Rewrite the integral in terms of the three variables.
- Evaluate the innermost integral first, then work your way outwards.
- Check your solution by plugging in the limits of integration and simplifying.
How do I know which order to integrate in?
The order of integration for a triple integral can be determined by looking at the given function and the limits of integration. It is generally easier to integrate in the order of dz, dy, dx or dx, dy, dz, depending on which variable is easiest to integrate first.
What is the purpose of using a triple integral?
A triple integral is used to calculate the volume under a surface in three-dimensional space. It is also used in many applications of physics and engineering, such as calculating the mass, center of mass, and moment of inertia of a three-dimensional object.
Can I change the order of integration?
Yes, the order of integration can be changed as long as the new limits of integration are equivalent to the original ones. However, some integrals may be easier to solve in a specific order, so it is important to choose the most efficient order for the given function.
How do I handle triple integrals with non-constant limits?
If the limits of integration are not constant, the integral can be rewritten as a double or single integral by using the properties of integration. Alternatively, the integral can be solved by breaking it up into multiple integrals, each with a different set of limits, and then adding them together.