# Integrating over unit circle

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

## Homework Equations

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

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

## 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:

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

## Homework Equations

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

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

## 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?

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

AB

vela
Staff Emeritus