Compute R |z|2dz, where γ is the straight line segment from 1 to 3 + 2i. γ

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.

Compute ∫ |z|² dz, where γ is the straight line segment from 1 to 3 + 2i.

Explore the solution to the line integral ∫ |z|² dz along the straight line from 1 to 3 + 2i. This detailed step-by-step explanation covers parametrization of complex curves, calculation of |z|², and splitting the integral into real and imaginary parts. Ideal for students studying complex analysis.

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