Integrating over unit circle

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Homework Statement



Express

f(x,y) = 1/sqrt(x^2 + y^2) . (y/sqrt(x^2 + y^2)) .exp(-2sqrt(x^2 + y^2))

in terms of polar coordinates [tex]\rho[/tex] and [tex]\varphi[/tex] then evaluate the integral over a circle of radius 1, centered at the origin.

Homework Equations



x = [tex]\rho[/tex]cos[tex]\varphi[/tex]
y = [tex]\rho[/tex]sin[tex]\varphi[/tex]

sin^2[tex]\varphi[/tex] + cos^2[tex]\varphi[/tex] = 1

The Attempt at a Solution



ok so here's my effort

after rearranging and substituting: f([tex]\rho[/tex],[tex]\varphi[/tex]) = sin[tex]\varphi[/tex]exp(-2[tex]\rho[/tex])

now let's integrate!
limits are 0 [tex]\leq[/tex] [tex]\rho[/tex] [tex]\leq[/tex]1
and 0 [tex]\leq[/tex] [tex]\varphi[/tex] [tex]\leq[/tex] 2[tex]\pi[/tex]

[tex]\int[/tex][tex]\int[/tex] sin[tex]\varphi[/tex]exp(-2[tex]\rho[/tex]) d(fi) d(rho)

the problem is sin becomes -cos so, -cos(2pi) - -cos(0) = 0

giving a final answer of zero doesn't make much sense, does it? so what arent i getting?
 
Last edited:

Answers and Replies

  • #2
660
0

Homework Statement



Express

f(x,y) = 1/sqrt(x^2 + y^2) . (y/sqrt(x^2 + y^2)) .exp(-2sqrt(x^2 + y^2))

in terms of polar coordinates [tex]\rho[/tex] and [tex]\varphi[/tex] then evaluate the integral over a circle of radius 1, centered at the origin.

Homework Equations



x = [tex]\rho[/tex]cos[tex]\varphi[/tex]
y = [tex]\rho[/tex]sin[tex]\varphi[/tex]

sin^2[tex]\varphi[/tex] + cos^2[tex]\varphi[/tex] = 1

The Attempt at a Solution



ok so here's my effort

after rearranging and substituting: f([tex]\rho[/tex],[tex]\varphi[/tex]) = sin[tex]\varphi[/tex]exp(-2[tex]\rho[/tex])

now let's integrate!
limits are 0 [tex]\leq[/tex] [tex]\rho[/tex] [tex]\leq[/tex]1
and 0 [tex]\leq[/tex] [tex]\varphi[/tex] [tex]\leq[/tex] 2[tex]\pi[/tex]

[tex]\int[/tex][tex]\int[/tex] sin[tex]\varphi[/tex]exp(-2[tex]\rho[/tex]) d(fi) d(rho)

the problem is sin becomes -cos so, -cos(2pi) - -cos(0) = 0

giving a final answer of zero doesn't make much sense, does it? so what arent i getting?

Didn't you drop a [tex]\rho[/tex] in the integrand?

AB
 
  • #3
vela
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You made a mistake when converting f to polar coordinates, and you made another one in writing down the integral. It turns out they cancel each other, so you got the right answer, which is 0.

Note that the original integrand is an odd function of y. Since the unit circle is symmetric about the y-axis, the integral turns out to be 0.
 

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