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.