Verified answer

Let x = 2 cos θ, y = √3 sin θ. (It's easy to see that the equation reduces to cos^2 θ + sin^2 θ = 1, so that this is a valid parameterisation of the solution.) We want to find the arc length as θ goes from 0 to π. This is just

∫(0 to π) √[(dx/dθ)^2 + (dy/dθ)^2] dθ

= ∫(0 to π) √(4 sin^2 θ + 3 cos^2 θ) dθ

= ∫(0 to π) √(sin^2 θ + 3) dθ

This is an elliptic integral of the second kind (they're called elliptic integrals because they arise from this very problem - determining the circumference of an ellipse), so there's no simple way to express the answer. As per the link below it's 5.87 to 3 s.f.

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