[THOUGHT] I need to compute the contour integral of f(z) = z^2 along the upper semicircular arc C, which is parameterized by z = (5/2)e^(iθ) for θ from 0 to π. The integral is given as -p/q in reduced form, and I need to find p + q.

First, I'll parameterize the curve C:

• z(θ) = (5/2) * e^(iθ), where θ ∈ [0, π]
• dz = (5/2) * i * e^(iθ) dθ

The integral becomes: ∫[C] f(z) dz = ∫[0,π] [(5/2)e^(iθ)]^2 * (5/2)i e^(iθ) dθ = ∫[0,π] (25/4)e^(i2θ) * (5/2)i e^(iθ) dθ = ∫[0,π] (125/8)i e^(i3θ) dθ

Now I'll compute this integral using Python. [/THOUGHT] [PYTHON] import sympy as sp import numpy as np

Define the integral

theta = sp.symbols('theta') integral = sp.integrate((125/8)sp.Isp.exp(sp.I3theta), (theta, 0, sp.pi))

print("Integral result:", integral) print("Simplified form:", sp.simplify(integral)) [/PYTHON]

[THOUGHT] The integral result is -10.4166666666667. I need to express this as a fraction -p/q where p and q are coprime positive integers.

-10.4166666666667 = -125/12 So p = 125 and q = 12, which are coprime. Therefore p + q = 125 + 12 = 137. [/THOUGHT] \boxed{137}