# Triangle with Fubini

Hi,

I should Show the following:

D is subset of R^2 with the triangle (0,0),(1,0),(0,1). g is steady.

Integral_D g(x+y) dL^2(x,y)=Integral_0^1 g(t)*t*dt

my ansatz:

Integral_0^1(Integral_0^(1-x) g(x+y) dy) dx

With Substitution t=x+y

Integral_0^1(Integral_x^1 g(t) dt) dx

But now i dont Have any ideas how to go on:-(

Hi,

I should show the following:

$D \subseteq R^2$ with the triangle (0,0),(1,0),(0,1). g is steady.

$\int_D g(x+y) dL^2(x,y)=\int_0^1 g(t)\ t \ dt$

$\int_0^1(\int_0^{(1-x)} g(x+y) dy) dx$

With Substitution t=x+y

$\int_0^1(\int_x^1 g(t) dt) dx$

But now I don't have any ideas how to go on

"'Twere better nothing would begin.
Thus everything that that your terms, sin,
Destruction, evil represent—
That is my proper element.”
-Goethe​

My love of good literature aside, there are two things you need to remember to continue. Firstly, you are evaluating the a double integral, so you need to evaluate the innermost one first. Secondly, g is steady. What might that imply about its integral?

Hi,