We find the total curvature of the portion of a smooth curve that runs from s = s₀ to s = s₁ > s₀ by integrating κ from s₀ to s₁. If the curve has some other parameter, say t, then the total curvature is

where t₀ and t₁ correspond to s₀ and s₁. Find the total curvatures of

a. The portion of the helix r(t) = (3 cos t)i + (3 sin t)j + tk, 0 ≤ t ≤ 4π.

b. The parabola y = x², -q

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