Integral Question: Swapping X and Y

[ehrenfest]

Homework Statement

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,

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

Homework Equations

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.

 

[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)?

 

[ehrenfest]

That makes sense.