 Since formula_39 and formula_40 the length of a quarter of the unit circle is

The 15-point Gauss–Kronrod rule estimate for this integral of differs from the true length of

by and the 16-point Gaussian quadrature rule estimate of differs from the true length by only .This means it is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations.Let formula_43 be a surface mapping and let formula_44 be a curve on this surface.