To compute the line integral $\int_{\gamma} |z|^2 \, dz$, where $\gamma$ is the straight line segment from $1$ to $3 + 2i$, we will follow these steps:

Step 1: Parametrize the line segment

The complex line segment from $1$ to $3 + 2i$ can be parametrized as:

$z(t) = 1 + t(3 + 2i - 1) = 1 + t(2 + 2i), \quad t \in [0, 1]$

Thus,

$z(t) = 1 + 2t + 2it = (1 + 2t) + 2it$

The derivative of $z(t)$ with respect to $t$ is:

$\frac{dz}{dt} = 2 + 2i$

Step 2: Express $|z(t)|^2$

We now compute $|z(t)|^2$. For a complex number $z = x + iy$, $|z|^2 = x^2 + y^2$.

For $z(t) = (1 + 2t) + 2it$, the real part $x(t) = 1 + 2t$ and the imaginary part $y(t) = 2t$. Therefore:

$|z(t)|^2 = (1 + 2t)^2 + (2t)^2$

Expanding both terms:

$|z(t)|^2 = (1 + 4t + 4t^2) + 4t^2 = 1 + 4t + 8t^2$

Step 3: Set up the integral

We can now set up the integral:

$\int_{\gamma} |z|^2 \, dz = \int_0^1 |z(t)|^2 \frac{dz}{dt} \, dt$

Substitute $|z(t)|^2 = 1 + 4t + 8t^2$ and $\frac{dz}{dt} = 2 + 2i$:

$\int_0^1 (1 + 4t + 8t^2)(2 + 2i) \, dt$

Step 4: Break the integral into real and imaginary parts

We expand the integrand:

$(1 + 4t + 8t^2)(2 + 2i) = (1 + 4t + 8t^2)(2) + (1 + 4t + 8t^2)(2i)$

This gives two parts:

$(2 + 8t + 16t^2) + (2i + 8it + 16it^2)$

Now, split the integral into real and imaginary parts:

$\text{Real part: } \int_0^1 (2 + 8t + 16t^2) \, dt$ $\text{Imaginary part: } \int_0^1 (2 + 8t + 16t^2) \, i \, dt$

Step 5: Compute the integrals

Real part:

$\int_0^1 (2 + 8t + 16t^2) \, dt = \int_0^1 2 \, dt + \int_0^1 8t \, dt + \int_0^1 16t^2 \, dt$

[ = 2[1]_0^1 + 8 \left[\frac{t^2}{2}\right]_0^1 + 16 \left[\frac{t^3}{3}\right]_0^1 ]

$= 2 + 8 \times \frac{1}{2} + 16 \times \frac{1}{3} = 2 + 4 + \frac{16}{3} = \frac{34}{3}$

Imaginary part:

$\int_0^1 (2 + 8t + 16t^2) \, dt = i \left( \frac{34}{3} \right)$

Step 6: Final answer

The final answer is the sum of the real and imaginary parts:

$\int_{\gamma} |z|^2 \, dz = \frac{34}{3} + i \frac{34}{3} = \frac{34}{3}(1 + i)$

Thus, the integral is:

$\boxed{\frac{34}{3}(1 + i)}$

Would you like more details on any part of this process or have additional questions?

Here are some related questions:

1. What are some examples of line integrals in physics?
2. How does the parametrization of a curve affect the integral?
3. What is the geometric meaning of $|z|^2$ in this context?
4. Can line integrals be computed over non-straight paths?
5. How would the answer change if $\gamma$ were a different path?

Tip: When parametrizing complex paths, always verify the derivative with respect to $t$ carefully to ensure accurate results.