[THOUGHT] I need to compute the contour integral of $f(z) = z^2$ over the upper semicircular arc $C$. The contour $C$ is parameterized by $z = \frac{5}{2} e^{i\theta}$ for $0 \le \theta \le \pi$. To compute the integral, I will parameterize the contour and then evaluate the integral using the parameter $\theta$.

The integral over $C$ is given by:

Substituting $f(z) = z^2$, we get:

Simplifying the integrand:

Now, I will compute this integral using Python to ensure accuracy. [/THOUGHT] [PYTHON] import numpy as np from scipy.integrate import quad

Define the integrand

def integrand(theta): return (125/8) * 1j * np.exp(3j * theta)

Compute the integral over theta from 0 to pi

result, _ = quad(lambda theta: np.real(integrand(theta)), 0, np.pi) + 1j * quad(lambda theta: np.imag(integrand(theta)), 0, np.pi)

The result should be purely imaginary, so we simplify

p = abs(result.imag) q = 1

Since the result is purely imaginary, we can extract the coefficient

The exact value is (125/8) * i * integral of e^{3i$\theta$} from 0 to pi

The integral of e^{3i$\theta$} is (e^{3i$\pi$} - 1)/(3i) = (e^{i$\pi$} - 1)/(3i) = (-1 - 1)/(3i) = -2/(3i) = 2i/3

So the total integral is (125/8) * i * (2i/3) = (125/8) * (-2/3) = -125/12

Thus, the integral is -125/12, so p = 125, q = 12, and p + q = 137

But let's verify numerically

print(result) [/PYTHON]

[THOUGHT] The Python code encountered a TypeError because the function did not handle the complex multiplication correctly. Let me re-evaluate the integral analytically to avoid the error. The integral simplifies to:

Since $e^{3i\pi} = -1$, the result is:

Thus, $p = 125$ and $q = 12$, so $p + q = 137$. [/THOUGHT] $\boxed{137}$