Easy Integration: Finding Flux of a Sphere Surface | Double Integral Help

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In summary, the conversation is about finding the flux of a surface and performing an integration with given bounds. The person is struggling with using Cartesian coordinates and suggests using parametrization instead. However, they are unsure of the bounds for the integration and it seems that the integral may diverge. A website is suggested for help with the trigonometry involved in the problem.
  • #1
JaysFan31

Homework Statement


I want to perform the following integration:
double integral of [(x^2)-(y^2)+2] dxdy where the function is subjected to the bound (x^2)+(y^2) greater than or equal to 2.

I'm trying to find the flux of a surface of a sphere (x^2)+(y^2)+(z^2)=9.

Homework Equations


Nothing, just rules of integration.

The Attempt at a Solution


Using Cartesian coordinates seems far too difficult. I could show you my work, but it's messy and complicated.
If I use parametrisation, then I get
double integral of [(r^2)cos^2(t)-(r^2)sin^2(t)+2) rdrdt
what are the bounds though?
t is between 0 and 2pi I'm pretty sure, but what about r?
It seems difficult since x^2+y^2 is greater than or equal to 2. This means that r^2 is greater than or equal to 2. Thus, it seems sqrt(2) is a lower bound, but what would the upper bound be?

Does this make any sense?
 
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  • #2
For the trig part: Hint: http://www.sosmath.com/trig/Trig5/trig5/trig5.html
However, considering the bound you stated, it looks like your integral diverges.
If you integrate only where x>2 and y<1, you already get divergence.
So, to answer your question: No.
 
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1. How do I find the flux of a sphere surface using double integration?

To find the flux of a sphere surface using double integration, you will need to first set up an integral using the surface integral formula. This formula involves the dot product between the vector field and the unit normal vector of the surface. Then, you will need to set up a double integral to integrate over the surface of the sphere. Finally, you can use the appropriate limits of integration and solve the integral to find the flux.

2. What is the unit normal vector of a sphere?

The unit normal vector of a sphere is a vector that is perpendicular to the surface of the sphere at any given point. It is also known as the surface normal vector. The direction of the unit normal vector is towards the outside of the sphere, and its magnitude is always equal to 1.

3. Can I use any vector field to find the flux of a sphere?

Yes, you can use any vector field to find the flux of a sphere. However, the vector field must be continuous and defined over the entire surface of the sphere. Additionally, the vector field should also be tangent to the surface of the sphere at any given point.

4. What are the limits of integration for finding the flux of a sphere?

The limits of integration for finding the flux of a sphere depend on the orientation of the sphere and the chosen coordinate system. For example, if the sphere is centered at the origin and has a radius of 1, the limits of integration for a double integral in spherical coordinates would be: ρ = 0 to ρ = 1, θ = 0 to θ = 2π, and φ = 0 to φ = π.

5. Can I use triple integration to find the flux of a sphere?

Yes, you can use triple integration to find the flux of a sphere. However, it is not necessary as the flux of a sphere can be found using a double integral. Triple integration would only be necessary if the vector field is changing in all three dimensions and cannot be simplified to a surface integral.

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