Dear all :
I need to solve the following integral (attached as an image file)
The context is on the calculation of View Factors in Radiation Heat Transfer
I worked out the this expression from the general formula, working out my configuration of two bodies in cylindrical coordinates (for one of the bodies) and using spherical coordinates (for the other body).
But I'm not sure if I ended with a well defined Integral, since I used two different coordinated systems on a same problem.
I used Cylindrical Coordinates for one of the dA and Spherical Coordinates for the other dA, however both dA are part of the same integral.
Hopefully someone out there can give me some help !
Regards and Thank you !
When is an integral well defined? First, if the integrating function is integrable (in this case it is); second, if the domain of integration makes sense and is non-contradictory, then the final answer is that it depends on domains A1 and A2. What I do not understand is what the two additional integration variables mean, therefore I suppose that in the domains A1 and A2 the limits for z and η are included. If not, it is a meaningless integral.
Hello Franklin Uriel Parás Hernández ,
Can you, please, more completely describe the system to the readers of this question? For example, is the spherical surface inside the cylindrical surface or the other way around? It is hard to tell from the way you posed your question.
Regards,
Tom Cuff
Dear Franklin Uriel Parás Hernández
Obviously one can try so calculate the integral "as it is seen" without geometrical interpretation. But the the following issues should be clear:
1. what are the meaning of the variables of integration; the first look suggests that 1.1. \phi and \theta are the sherical angles - are they varying within the full range of (-pi; \pi] and [-pi/2 ; \pi/2] or are they restricted to some part of the shere?
1.2. z and \eta are the length coordinate and \eta the angle arround the axis; again what is their range of variability?
2. The range of the two types of variables must be different (A1 cannot equal A2).
3. If the above meaning of the variables in point 1 above is consistent with your choice, then the integral is a product of the integral with respect to \phi and \theta over A1 and the area of the cylindrical surface divided by the radius (since the integrant is independent of z and eta).
Hopefully, this helps. Best, Joachim
Dear Franklin Uriel Parás Hernández . This formula may help: sin(2a)cos(b)=[sin(2a+b)+sin(2a-b)]/2
The integral wrt the first two coordinates over the whole sphere is obviously zero. THEREFORE the answer by the Author about the region of integration is of greatest importance.
Joachim
Obviously, the complexity of this integral lies entirely in the domains of integration. If these are fixed intervals of θ, φ, z, and η, the interval is trivial to work out. However, if, say, the integral on θ is done before that on φ, and we have integration limits θ1(φ) and θ2(φ) , then the subsequent φ integration can be arbitrarily difficult and may be analytically impossible, depending on the functions θ1(φ) and θ2(φ). A similar remark can be offered for the variables z and η. Therefore, I agree with the statement made before that it is crucial to properly indicate the boundaries of integration.
No it is not well-defined integral.
There are two hurdles?
1) Repeated domain.
2) Two extra integrands at the end of the expression.
Accordingly it is absurd.
When is an integral well defined? First, if the integrating function is integrable (in this case it is); second, if the domain of integration makes sense and is non-contradictory, then the final answer is that it depends on domains A1 and A2. What I do not understand is what the two additional integration variables mean, therefore I suppose that in the domains A1 and A2 the limits for z and η are included. If not, it is a meaningless integral.
It is to be, provided A1 is explicit in the first two variables and A2 is explicit in the last two variables. Under these conditions the solution is assured
The end product of the integration illustrates
- A1-sphere (subsphere) of radius equal to half of any A2-variable placed on A2-square plane
- A1-sphere (subsphere) of radius equal to half of A2-last variable placed on top of cylinder of cross sectional said radius and height A2-fist varable
- the process being continuous, A2-fist and A2-last variables are interchangeable
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