Question

Let \( \mathbf{F}(x, y, z) = 2y \hat{i} + x^2 \hat{j} + xy \hat{k} \) and let \( C \) be the curve of intersection of the plane

\[ x + y + z = 1 \]

and the cylinder

\[ x^2 + y^2 = 1. \]

Then the value of

\[ \left| \int_C \mathbf{F} \cdot d\mathbf{r} \right| \]

• A. \( \pi \)
• B. \( \frac{3\pi}{2} \)
• C. \( 2\pi \)
• D. \( 3\pi \)

Hint:

For integrals along curves, parameterize the curve and compute the dot product of the vector field and the differential vector to evaluate the line integral.

Answer 1

The Correct Option is C

Step 1: Parameterize the curve.
The curve \( C \) is the intersection of the plane \( x + y + z = 1 \) and the cylinder \( x^2 + y^2 = 1 \). We can parameterize the curve using cylindrical coordinates: \[ x = \cos t, \quad y = \sin t, \quad z = 1 - \cos t - \sin t, \quad t \in [0, 2\pi]. \]
Step 2: Compute the vector field along the curve.
The vector field \( \mathbf{F}(x, y, z) = 2y \hat{i} + x^2 \hat{j} + xy \hat{k} \) along the curve is: \[ \mathbf{F}(x(t), y(t), z(t)) = 2\sin t \hat{i} + \cos^2 t \hat{j} + \cos t \sin t \hat{k}. \]
Step 3: Compute the line integral.
The line integral is: \[ \int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} \left( 2\sin t \, \frac{d}{dt}(\cos t) + \cos^2 t \, \frac{d}{dt}(\sin t) + \cos t \sin t \, \frac{d}{dt}(1 - \cos t - \sin t) \right) dt. \] After performing the integration, we find that the value of the integral is \( 2\pi \).

Step 4: Conclusion.
Thus, the correct answer is \( \boxed{(C)} \).