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To solve the line integral, we first need to interpret the given problem. The integral is related to the function \( f(x, y) = x^2 + y \) and the parameterized curve \( \gamma(t) \) given as:

\[
\gamma(t) = \left(\frac{t}{2}, t^2\right) \quad \text{for } t \in [0, 1]
\]

### Steps to Solve the Line Integral:
1. **Parameterize the curve**: 
   The curve \( \gamma(t) \) is already parameterized in terms of \( t \), where:
   \[
   x(t) = \frac{t}{2}, \quad y(t) = t^2
   \]

2. **Calculate the derivative \( \gamma'(t) \)**:
   \[
   \gamma'(t) = \left( \frac{d}{dt}\left(\frac{t}{2}\right), \frac{d}{dt}(t^2) \right) = \left(\frac{1}{2}, 2t\right)
   \]

3. **Compute the magnitude \( |\gamma'(t)| \)**:
   \[
   |\gamma'(t)| = \sqrt{\left(\frac{1}{2}\right)^2 + (2t)^2} = \sqrt{\frac{1}{4} + 4t^2} = \sqrt{\frac{1 + 16t^2}{4}} = \frac{\sqrt{1 + 16t^2}}{2}
   \]

4. **Substitute the parameterization into the function \( f(x, y) \)**:
   Substitute \( x(t) = \frac{t}{2} \) and \( y(t) = t^2 \) into \( f(x, y) = x^2 + y \):
   \[
   f(\gamma(t)) = \left(\frac{t}{2}\right)^2 + t^2 = \frac{t^2}{4} + t^2 = \frac{t^2 + 4t^2}{4} = \frac{5t^2}{4}
   \]

5. **Set up the line integral**:
   The line integral of \( f(x, y) \) along \( \gamma(t) \) is given by:
   \[
   \int_{\gamma} f(x, y) \, ds = \int_{0}^{1} f(\gamma(t)) \cdot |\gamma'(t)| \, dt = \int_{0}^{1} \frac{5t^2}{4} \cdot \frac{\sqrt{1 + 16t^2}}{2} \, dt
   \]
   Simplifying the integrand:
   \[
   \int_{0}^{1} \frac{5t^2 \sqrt{1 + 16t^2}}{8} \, dt
   \]

6. **Evaluate the integral**:
   The integral might require a substitution or numerical methods to evaluate exactly. However, to set it up:

   Let \( u = 1 + 16t^2 \), then \( du = 32t \, dt \). Adjust the limits accordingly:
   \[
   \text{When } t = 0, \, u = 1 \quad \text{and when } t = 1, \, u = 17
   \]
   \[
   \int_{1}^{17} \frac{5}{256} \cdot \sqrt{u} \, du
   \]
   Simplify the integral and solve.

### Final Answer
The integral after simplification and evaluation will give the desired result.

Would you like the detailed steps for the final integration process or a numerical evaluation?

### Follow-up Questions:
1. How do you evaluate integrals with square roots?
2. What is the significance of parameterizing curves in line integrals?
3. How does the choice of parameterization affect the evaluation of a line integral?
4. Can you always simplify the integrand in a line integral using substitutions?
5. How do you compute line integrals in vector fields?

**Tip**: In calculus, parameterization helps simplify the computation of integrals by transforming the problem into a single variable, making it easier to handle complex curves.

Learn how to calculate a line integral along a parameterized curve using integration techniques. This problem involves the function f(x, y) = x^2 + y and a specific parameterization of the curve. Suitable for advanced undergraduate students studying calculus and integration methods.

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