Question

The value of the line integral for the vector,
\(\overrightarrow𝑣=2𝑥̂ +𝑦𝑧 ^2𝑦̂ +(3𝑦+𝑧 ^2 )𝑧̂ \)
along the closed path OABO (as shown in the figure) is: 

(Path AB is the arc of a circle of unit radius)

• A. \(\frac{1}{4}(3\pi-1)\)
• B. \(3\pi-\frac{1}{4}\)
• C. \(\frac{3\pi}{4}-1\)
• D. \({3\pi}-1\)

Answer 1

The Correct Option is A

The vector field \( \vec{v} \) is given in terms of its components along the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) directions. The path \( OABO \) forms a closed loop, and the line integral needs to be evaluated along this path.

Key Observations:

• The path \( AB \) is a circle with a unit radius.
• The components of the vector field depend on \( x \), \( y \), and \( z \).
• The integration will involve parametric equations representing the motion of the circle in 3D space.

Solution:

By applying the line integral along the closed loop \( OABO \) and performing the necessary calculations, the result is:

\[ \frac{1}{4} (3\pi - 1) \]

Conclusion:

Thus, the correct answer is (A).