Question

The value of \( \oint_S \vec{F} \cdot d\vec{s} \) where \( \vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k} \) taken over the cylinder \( x^2+y^2=4, z=0 \) and \( z=3 \) is:

• A. 126\(\pi\)
• B. 168\(\pi\)
• C. 42\(\pi\)
• D. 84\(\pi\)

Hint:

When asked to evaluate a surface integral over a simple closed surface (like a sphere, cylinder, or cube), always check if the Divergence Theorem can be applied first. It often simplifies the problem from multiple surface integrals to a single, often easier, volume integral. Also, look for symmetries that might make parts of the integral zero, like integrating \(y\) or \(sin\theta\) over a symmetric domain.

Answer 1

The Correct Option is D

Step 1: Conceptualization:
The objective is to compute the surface integral of a vector field \( \vec{F} \) across a closed cylindrical surface. This integral represents the net flux of \( \vec{F} \) exiting the surface. The Divergence Theorem is applicable here, transforming the surface integral into a volume integral of the vector field's divergence.Step 2: Methodological Framework:
The Divergence Theorem is stated as:\[ \oint_S \vec{F} \cdot d\vec{s} = \iiint_V (abla \cdot \vec{F}) \, dV \]where V denotes the volume enclosed by the surface S. The initial step is to compute the divergence of \( \vec{F} \).\[ abla \cdot \vec{F} = \frac{\partial}{\partial x}(4x) + \frac{\partial}{\partial y}(-2y^2) + \frac{\partial}{\partial z}(z^2) \]Step 3: Procedural Breakdown:
1. Divergence Calculation:
\[ abla \cdot \vec{F} = 4 - 4y + 2z \]2. Volume Integral Setup:
The region of integration V is a cylinder with radius \(r=2\) and height extending from \(z=0\) to \(z=3\). The integral is formulated as:\[ \iiint_V (4 - 4y + 2z) \, dV \]This integral is best evaluated using cylindrical coordinates. The transformation equations are \( x = r\cos\theta, y = r\sin\theta, z = z \), and the volume element is \( dV = r \, dz \, dr \, d\theta \). The integration limits are \( 0 \le \theta \le 2\pi \), \( 0 \le r \le 2 \), and \( 0 \le z \le 3 \). The integral in cylindrical coordinates becomes:\[ \int_0^{2\pi} \int_0^2 \int_0^3 (4 - 4r\sin\theta + 2z) \, r \, dz \, dr \, d\theta \]3. Integral Evaluation:
The integral can be decomposed into three separate integrals:\[ \int_0^{2\pi} \int_0^2 \int_0^3 4r \, dz \, dr \, d\theta - \int_0^{2\pi} \int_0^2 \int_0^3 4r^2\sin\theta \, dz \, dr \, d\theta + \int_0^{2\pi} \int_0^2 \int_0^3 2zr \, dz \, dr \, d\theta \]Component 1: \( \int_0^{2\pi} d\theta \int_0^2 r dr \int_0^3 4 dz = (2\pi) \times [\frac{r^2}{2}]_0^2 \times [4z]_0^3 = 2\pi \times 2 \times 12 = 48\pi \).Component 2: The integral \( \int_0^{2\pi} \sin\theta \, d\theta = [-\cos\theta]_0^{2\pi} = 0 \). Consequently, this entire component evaluates to 0.Component 3: \( \int_0^{2\pi} d\theta \int_0^2 r dr \int_0^3 2z dz = (2\pi) \times [\frac{r^2}{2}]_0^2 \times [z^2]_0^3 = 2\pi \times 2 \times 9 = 36\pi \).Total Flux:The aggregate value of the integral is the sum of the computed components:\[ 48\pi - 0 + 36\pi = 84\pi \]Step 4: Conclusion:
The computed value of the surface integral is \(84\pi\).