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For $x,y \in \mathbb{R}$, function $f(x,y)$ is defined as $$f(x,y) = 1 \quad\textrm{if}\quad x=y$$ $$f(x,y) = 0 \quad\textrm{if}\quad x\neq y$$

It seems to me that the integral $I = \int_0^1 \int_0^1 f(x,y) dxdy$ should equal to the diagonal of the unit square, hence $$I = \int_0^1 \int_0^1 f(x,y) dxdy = \sqrt{2} \tag{1}$$

However, in writing the integral mathematically, I got $$I = \int_0^1 \int_0^1 f(x,y) dxdy= \int_0^1 1 dx = 1 \tag{2}$$

Could anyone please show me where I was wrong in (1) and/or (2)?

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Not sure how you think in (1) but (2) is the correct answer.

Imagine the x- and y-axes and that you start in the origin (0,0). Now do the integral of your function over dx i.e. move over the x-axis from (0,0) to (1,0). The only time your integral is not zero is when x=y=0, you get that the first integral is equal to 1 and you have ended up at the point (1,0). Now do the integral over dy. You move from (1,0) to (1,1) and your integral will only be non-zero were x=y=1 and the second integral is thus also equal to 1 and the entire two dimensional integral is therefore one.

Hope this helps!

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The double integral represents the area of the diagonal line, not the length. This is $0$.

When doing the innermost integral, $\int_0^1f(x,y)dx$, the $y$ is considered a fixed number, so the integrand is $0$ everywhere except at a single point, where it is $1$. This integral becomes $0$, so when you do the outer integral, you get $\int_0^10\,dy$.

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What you have is the ordinary function $[x = y]$, which is not the same as $\delta(x - y)$; the latter isn't an ordinary function at all. The double integral of $[x = y]$ is trivially zero.

If you adopt some natural assumptions about how $\delta(f)$ is defined, then the formula giving the length of a curve cannot be just $\iint \delta(f(x, y)) \, dx dy$, see this answer. The formula you get is $$l = \int_0^1 \int_0^1 \delta(x - y) \, |\nabla (x - y)| \, dx dy,$$ which is then equal to $\int_0^1 \sqrt 2 \, dy$.

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