# Integrating over unit circle

1. Jan 25, 2010

### iloveannaw

1. The problem statement, all variables and given/known data

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 $$\rho$$ and $$\varphi$$ then evaluate the integral over a circle of radius 1, centered at the origin.

2. Relevant equations

x = $$\rho$$cos$$\varphi$$
y = $$\rho$$sin$$\varphi$$

sin^2$$\varphi$$ + cos^2$$\varphi$$ = 1

3. The attempt at a solution

ok so here's my effort

after rearranging and substituting: f($$\rho$$,$$\varphi$$) = sin$$\varphi$$exp(-2$$\rho$$)

now let's integrate!
limits are 0 $$\leq$$ $$\rho$$ $$\leq$$1
and 0 $$\leq$$ $$\varphi$$ $$\leq$$ 2$$\pi$$

$$\int$$$$\int$$ sin$$\varphi$$exp(-2$$\rho$$) 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: Jan 25, 2010
2. Jan 25, 2010

### Altabeh

Didn't you drop a $$\rho$$ in the integrand?

AB

3. Jan 25, 2010

### vela

Staff Emeritus
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.