# Arc length In $$L_\infty$$ Metric

Calculus Level 5

The $$L_\infty$$ metric defined on the Euclidean plane is a metric where the distance between two points $$(a, b)$$ and $$(x, y)$$ is $$\max \{|x-a|, |y-b|\}$$.

Consider the arc $$(t, t^2)$$ for $$t \in [-1, 1]$$. What is the length of this arc in the $$L_\infty$$ metric?

This problem is obviously inspired from the $$L_1$$ metric.

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