CS
  can be parametrized by r(s,t)=⟨
 
s
functionsequation editor , 
sin(t)
functionsequation editor ,1cos(t)⟩
 with 0≤s≤4
 and 0≤t≤2π
.

∬CSF⋅dCS=∫2π0∫40
 
functionsequation editor ds dt

To evaluate the given surface integral, we start with the parameterized surface:

\[
r(s, t) = \langle s, \sin(t), \cos(t) \rangle \quad \text{with} \quad 0 \leq s \leq 4 \quad \text{and} \quad 0 \leq t \leq 2\pi.
\]

The goal is to compute:

\[
\iint_{C_S} \mathbf{F} \cdot d\mathbf{S} = \int_0^{2\pi} \int_0^4 \mathbf{F} \cdot (\mathbf{r}_s \times \mathbf{r}_t) \, ds \, dt.
\]

### Step 1: Compute the partial derivatives
The partial derivatives of \( r(s, t) \) with respect to \( s \) and \( t \) are:
\[
\mathbf{r}_s = \frac{\partial \mathbf{r}}{\partial s} = \langle 1, 0, 0 \rangle, \quad \mathbf{r}_t = \frac{\partial \mathbf{r}}{\partial t} = \langle 0, \cos(t), -\sin(t) \rangle.
\]

### Step 2: Compute the cross product
The cross product \( \mathbf{r}_s \times \mathbf{r}_t \) is:
\[
\mathbf{r}_s \times \mathbf{r}_t = 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 0 & 0 \\
0 & \cos(t) & -\sin(t)
\end{vmatrix}.
\]

Expanding this determinant:
\[
\mathbf{r}_s \times \mathbf{r}_t = \mathbf{i}(0 - 0) - \mathbf{j}(0 - 0) + \mathbf{k}(1 \cdot \cos(t) - 0),
\]
\[
\mathbf{r}_s \times \mathbf{r}_t = \langle 0, 0, \cos(t) \rangle.
\]

### Step 3: Set up the integral
Assuming \( \mathbf{F} \) is a vector field, we need to evaluate:
\[
\mathbf{F} \cdot (\mathbf{r}_s \times \mathbf{r}_t) = \mathbf{F} \cdot \langle 0, 0, \cos(t) \rangle.
\]

Let \( \mathbf{F}(s, t) \) be given explicitly or depend on \( s, t \). Substitute the appropriate expression for \( \mathbf{F} \) to proceed.

If \( \mathbf{F} = \langle F_1, F_2, F_3 \rangle \), then:
\[
\mathbf{F} \cdot (\mathbf{r}_s \times \mathbf{r}_t) = F_3(s, t) \cos(t).
\]

The integral becomes:
\[
\int_0^{2\pi} \int_0^4 F_3(s, t) \cos(t) \, ds \, dt.
\]

### Step 4: Evaluate the integral
- If \( F_3(s, t) \) is given explicitly, substitute it into the integral and calculate.

Would you like to provide additional information about \( \mathbf{F} \) or continue with further general assumptions? 

---

**Questions to expand understanding:**
1. What is the explicit form of the vector field \( \mathbf{F} \)?
2. How is the cross product \( \mathbf{r}_s \times \mathbf{r}_t \) used in surface integrals generally?
3. What is the geometric meaning of \( \mathbf{r}_s \times \mathbf{r}_t \) in terms of the surface?
4. How would this change if the bounds for \( s \) or \( t \) varied?
5. What simplifications occur if \( \mathbf{F} \) has specific symmetry?

**Tip:** Always verify the parameterization of the surface before computing integrals, as errors here propagate through the calculation.

CS can be parametrized by r(s,t) = ⟨s, sin(t), cos(t)⟩ with 0 ≤ s ≤ 4 and 0 ≤ t ≤ 2π. ∬CSF⋅dCS = ∫₀²π∫₀⁴ ds dt

Explore how to evaluate the surface integral of a parametrized surface r(s,t) = ⟨s, sin(t), cos(t)⟩ with 0 ≤ s ≤ 4 and 0 ≤ t ≤ 2π. This detailed solution demonstrates vector calculus techniques, including surface parameterization, cross products, and setting up the integral. Suitable for Calculus III students.

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