This can be found in usual way of calculus. First, we find the parametrization of the space curve under consideration:

sol := solve({z = x^2+2*y^2, z = -2*x^2-y^2+3}, {x, y});

{x = RootOf(_Z^2+z-2), y = RootOf(_Z^2-z+1)}

Next,

allvalues(sol);

{x = sqrt(-z+2), y = sqrt(z-1)}, {x = sqrt(-z+2), y = -sqrt(z-1)}, {x = -sqrt(-z+2), y = sqrt(z-1)}, {x = -sqrt(-z+2), y = -sqrt(z-1)}

Now the variables x and y are expressed through z in the interval from 1 to 2. It should be noted that the curve splits in the four parametrized curves having the same length.

We plot it by

A := plots:-spacecurve([sqrt(-z+2), sqrt(z-1), z], z = 1 .. 2); B := plots:-spacecurve([sqrt(-z+2), -sqrt(z-1), z], z = 1 .. 2); C := plots:-spacecurve([-sqrt(-z+2), sqrt(z-1), z], z = 1 .. 2); E := plots:-spacecurve([-sqrt(-z+2), -sqrt(z-1), z], z = 1 .. 2);

plots:-display([A, B, C, E], axes = frame);

Second, we find its length numerically:

VectorCalculus:-ArcLength(z -><sqrt(-z+2), sqrt(z-1), z>, 1 .. 2, inert); 4*evalf(%);

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