Ⓒ Evaluate the line integral √ √xy dx + e ³ dy + xz dz с where C is panemetrvied by r(t)= , 0£t£l.

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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This is Calculus 3
**Line Integrals and Parameterization**

**Problem Statement:**
Evaluate the line integral 

\[ \int_{C} \sqrt{xy} \, dx + e^{-y} \, dy + xz \, dz, \]

where \(C\) is parameterized by \( \mathbf{r}(t) = \langle t^4, t^2, t^3 \rangle \), with \(0 \leq t \leq 1\).

**Explanation:**

1. **Function to Integrate:**
   The integrand is composed of three parts: \( \sqrt{xy} \, dx \), \( e^{-y} \, dy \), and \( xz \, dz \).

2. **Parameterization of the Curve \(C\):**
   The curve \(C\) is given by the parameterization vector function:
   \[ \mathbf{r}(t) = \langle t^4, t^2, t^3 \rangle, \]
   for \( t \) in the interval \([0, 1]\).

3. **Components of Parameterization:**
   - \( x(t) = t^4 \)
   - \( y(t) = t^2 \)
   - \( z(t) = t^3 \)

4. **Differentials:**
   - \( dx = \frac{d(x(t))}{dt} \, dt = 4t^3 \, dt \)
   - \( dy = \frac{d(y(t))}{dt} \, dt = 2t \, dt \)
   - \( dz = \frac{d(z(t))}{dt} \, dt = 3t^2 \, dt \)

5. **Substitute Parameterization into Integrand:**
   Replace \( x \), \( y \), \( z \), \( dx \), \( dy \), and \( dz \) with their parameterized forms and compute the line integral over the given range for \( t \). 

This problem requires careful substitution and integration within the specified limits to find the value of the integral. The approach outlined provides the necessary steps to resolve the line integral given the parameterization \( \mathbf{r}(t) \).
Transcribed Image Text:**Line Integrals and Parameterization** **Problem Statement:** Evaluate the line integral \[ \int_{C} \sqrt{xy} \, dx + e^{-y} \, dy + xz \, dz, \] where \(C\) is parameterized by \( \mathbf{r}(t) = \langle t^4, t^2, t^3 \rangle \), with \(0 \leq t \leq 1\). **Explanation:** 1. **Function to Integrate:** The integrand is composed of three parts: \( \sqrt{xy} \, dx \), \( e^{-y} \, dy \), and \( xz \, dz \). 2. **Parameterization of the Curve \(C\):** The curve \(C\) is given by the parameterization vector function: \[ \mathbf{r}(t) = \langle t^4, t^2, t^3 \rangle, \] for \( t \) in the interval \([0, 1]\). 3. **Components of Parameterization:** - \( x(t) = t^4 \) - \( y(t) = t^2 \) - \( z(t) = t^3 \) 4. **Differentials:** - \( dx = \frac{d(x(t))}{dt} \, dt = 4t^3 \, dt \) - \( dy = \frac{d(y(t))}{dt} \, dt = 2t \, dt \) - \( dz = \frac{d(z(t))}{dt} \, dt = 3t^2 \, dt \) 5. **Substitute Parameterization into Integrand:** Replace \( x \), \( y \), \( z \), \( dx \), \( dy \), and \( dz \) with their parameterized forms and compute the line integral over the given range for \( t \). This problem requires careful substitution and integration within the specified limits to find the value of the integral. The approach outlined provides the necessary steps to resolve the line integral given the parameterization \( \mathbf{r}(t) \).
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