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To calculate the length of the given curves, we will use the formula for the length of a parametric curve \(\mathbf{r}(t) = (x(t), y(t), z(t))\), which is given by:

\[
L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt
\]

Let's analyze both curves individually.

### Curve 1:

Given:
\[
\mathbf{r}_1(t) = \begin{pmatrix} t \\ \frac{t}{2} \\ \frac{t^2}{2} \end{pmatrix}, \quad t \in [0, 2]
\]

1. **Derivatives**:
   - \( \frac{dx}{dt} = 1 \)
   - \( \frac{dy}{dt} = \frac{1}{2} \)
   - \( \frac{dz}{dt} = t \)

2. **Length Calculation**:
   - The integrand becomes:
   \[
   \sqrt{1^2 + \left(\frac{1}{2}\right)^2 + t^2} = \sqrt{1 + \frac{1}{4} + t^2} = \sqrt{\frac{5}{4} + t^2}
   \]
   - Therefore, the length is:
   \[
   L_1 = \int_{0}^{2} \sqrt{\frac{5}{4} + t^2} \, dt
   \]

### Curve 2:

Given:
\[
\mathbf{r}_2(t) = \begin{pmatrix} \sin(t) \\ \cos(t) \\ t \end{pmatrix}, \quad t \in [0, \frac{\pi}{2}]
\]

1. **Derivatives**:
   - \( \frac{dx}{dt} = \cos(t) \)
   - \( \frac{dy}{dt} = -\sin(t) \)
   - \( \frac{dz}{dt} = 1 \)

2. **Length Calculation**:
   - The integrand becomes:
   \[
   \sqrt{\cos^2(t) + (-\sin(t))^2 + 1^2} = \sqrt{\cos^2(t) + \sin^2(t) + 1} = \sqrt{1 + 1} = \sqrt{2}
   \]
   - Therefore, the length is:
   \[
   L_2 = \int_{0}^{\frac{\pi}{2}} \sqrt{2} \, dt = \sqrt{2} \cdot \frac{\pi}{2}
   \]

### Results:

- For the first curve: The integral \(L_1 = \int_{0}^{2} \sqrt{\frac{5}{4} + t^2} \, dt\) needs to be computed.
- For the second curve: \( L_2 = \frac{\pi \sqrt{2}}{2} \).

Would you like me to compute the integral for \( L_1 \) numerically or show the steps for exact evaluation?

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**Relative Questions:**

1. How would the curve length formula change for a curve defined in polar coordinates?
2. What is the geometric interpretation of the integrand in the curve length formula?
3. How does the curve length formula adapt for 2D curves?
4. What happens to the length if one of the component functions is a constant?
5. Can we derive the arc length of a circle using this method?

**Tip:** When working with parametric curves, always double-check the limits of integration, as they depend on the parameter interval.

Learn how to calculate the length of parametric curves using integral calculus. Step-by-step guide with examples including parametric equations and integrand calculations. Suitable for advanced undergraduate students.

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