Improper integral with spherical coordinates

  • #1
Cyn
8
0

Homework Statement


I have a question.

I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent.

x^2y^2z^2/r^(17/2) * f(x,y,z)dV.

Homework Equations



Sphericak coordinates
x = rsin(a)cos(b)
y = rsin(a)sin(b)
z = rcos(a)

The Attempt at a Solution



Because you know that f continuous and positive is can you say that the integral of f is between m and M. But now, I have to know what the other integral is. I have to use sphericak coordinates.
x = rsin(a)cos(b)
y = rsin(a)sin(b)
z = rcos(a)

The determinant is r^2sin(a). But if I calculate this with the standard boundaries:
r between 0 and R (R=1)
a between 0 and pi
b between -pi and pi

And if I take r between 1/m and 1 and let the limit m-->infinity, then I find that the integral is 8/105 pi. But the answer is 16/105 pi.
Have I do something wrong or have I need to use different boundaries?

Thank you
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement


I have a question.

I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent.

x^2y^2z^2/r^(17/2) * f(x,y,z)dV.

Homework Equations



Sphericak coordinates
x = rsin(a)cos(b)
y = rsin(a)sin(b)
z = rcos(a)

The Attempt at a Solution



Because you know that f continuous and positive is can you say that the integral of f is between m and M. But now, I have to know what the other integral is. I have to use sphericak coordinates.
x = rsin(a)cos(b)
y = rsin(a)sin(b)
z = rcos(a)

The determinant is r^2sin(a). But if I calculate this with the standard boundaries:
r between 0 and R (R=1)
a between 0 and pi
b between -pi and pi

And if I take r between 1/m and 1 and let the limit m-->infinity, then I find that the integral is 8/105 pi. But the answer is 16/105 pi.
Have I do something wrong or have I need to use different boundaries?

Thank you

Show your actual work; it looks like you might have made a simple error, but we cannot tell without seeing what you did. Also: you need to tell us what is the formula for the function f(x,y,z).

When you show your work, please do NOT post an image of handwritten material; take the time to type it out. (Note, for example, that you can type ##\int_a^b f(x) \, dx## as int_{x=a..b} f(x) dx in plain text, so it should not be too bad.)
 

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