[THOUGHT] To find the integral $\int_C f(z) \, dz$ where $f(z) = z^2$ and $C$ is the upper semicircular arc $z = \frac{5}{2} e^{i\theta}$ from $\theta = 0$ to $\theta = \pi$, we can use the parametric form of the integral. The integral can be written as:

First, compute the derivative:

Next, substitute $z = \frac{5}{2} e^{i\theta}$ into $f(z)$:

Now, substitute these into the integral:

Factor out the constant:

To solve $\int_0^\pi e^{3i\theta} \, d\theta$, use the antiderivative of $e^{3i\theta}$:

Evaluate this from $0$ to $\pi$:

Recall that $e^{3i\pi} = e^{i\pi} \cdot e^{i\pi} \cdot e^{i\pi} = (-1) \cdot (-1) \cdot (-1) = -1$:

Substitute this back into the integral:

Simplify $-\frac{250}{24}$:

Since $125$ and $12$ are relatively prime, we have $p = 125$ and $q = 12$. Thus, $p + q = 125 + 12 = 137$.

[/THOUGHT] [PYTHON]

Verify the simplification and calculation in Python

from fractions import Fraction

Numerator and denominator

numerator = 250 denominator = 24

Simplify the fraction

simplified_fraction = Fraction(numerator, denominator) p = simplified_fraction.numerator q = simplified_fraction.denominator

Calculate p + q

result = p + q print(result)

[THOUGHT] The Python output confirms that the simplified fraction is $-\frac{125}{12}$. Therefore, $p = 125$ and $q = 12$. The sum $p + q$ is indeed 137. [/THOUGHT] \boxed{137}