# Integral Question

1. Jul 30, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
Can someone explain to me the logic of this statement:

"Since the value of f (x, y) is unchanged when we swap x with y,

$$\int_0^1 \int_0^x f (x+y)dydx = 1/2 \int_0^1 \int_0^1 f (x+y)dydx.$$"

2. Relevant equations

3. The attempt at a solution

$$\int_0^1 \int_0^x f (x+y)dydx = \int_0^1 \int_0^y f (y+x)dxdy$$

But I do not think that is the same.

2. Jul 30, 2007

### Dick

Draw a picture of the region of integration. It's one half of the unit square. What is the integral over the other half equal to if f(x,y)=f(y,x)?

3. Jul 30, 2007

### ehrenfest

That makes sense.