Use of min(x,y) in plot3d, integrals, etc.

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Is there a way to get for the function `min(x,...,y)' the following?

1. a plot3d? (in the case of two arguments)
2. integrals?
3. at least, numerical integrals?

It's likely my bad syntax the culprit, but here is my MWE:

var('x,y,z') 
plot3d(min(y,x),(x,0,1),(y,0,1)) #not the right plot
integrate(integrate(integrate(min(y,x,z),(x,0,1)),(y,0,1)),(z,0,1)) # this integrates f(x)=x
numerical_integral(numerical_integral(numerical_integral(min(y,x,z),(x,0,1)),(y,0,1)),(z,0,1))