How would I go about computing the following function:
$$F(x_1,x_2) = \int_0^1 \chi_{[0,1]^2}(x_1 - t, x_2-t) \ dt$$
My idea is to observe that
$$F(x_1, x_2) = \int_0^1 \chi_{[0,1]} (x_1-t) \chi_{[0,1]} (x_2-t) dt$$
But then I'm unsure what to do afterwards.
Hint: $[x_1-1,x_1]\cap [x_2-1,x_2]$ is an interval (or empty). (You can write down the end points of this interval by considering various cases). $F(x_1,x_2)$ is the length of this interval (or $0$ if the intersection is empty) because the integrand is $1$ if $t$ is in this interval and $0$ otherwise.