(a) Evaluate ∫_C f(z) dz by Theorem 1 and check the result by Theorem 2, where:

(i) f(z) = z⁴ and C is the semicircle |z| = 2 from 2i to 2i in the right half-plane,

(ii) f(z) = e^2z and C is the shortest path from 0 to 1 + 2i.

(b) Experiment with a family of paths with common endpoints, say, z(t) = t + iα sin t, 0 ≤ t ≤ π with real parameter α. Integrate nonanalytic functions (Re z, Re (z²), etc.) and explore how the result depends on α. Then take analytic functions of your choice. (Show the details of your work.) Compare and comment.

(c) Choose another family, for example, semi-ellipses z(t) = α cos t + i sin t, -π/2 ≤ t ≤ π/2.