Arc Length In A Rectilinear Metric?

Calculus Level 4

The Minkowski plane is defined by a rectilinear metric as follows:

For points $(a, b)$ and $(m, n)$ , the distance between these points is

$d((a,b);(m,n))=|m-a|+|n-b|$

What is the length of the arc of $f(x)=x^2$ from $x=-1$ to $x=1$ in this metric?

Notation: $| \cdot |$ denotes the absolute value function.