Question

The value of the integral
\[ \oint_C \left( y^3 \mathbf{i} - x^3 \mathbf{j} \right) \cdot \left( i \, dx + j \, dy \right) \] where \(C\) is the closed curve, is:

• A. \(-\frac{3}{2} \pi a^3\)
• B. \(-\frac{3}{5} \pi a^4\)
• C. \(-\frac{5}{3} \pi a^4\)
• D. \(-\frac{3}{2} \pi a^3\)

Hint:

For closed line integrals use Green’s Theorem to convert the line integral into a double integral over the region enclosed by the curve

Answer 1

The Correct Option is A

Green's Theorem transforms the line integral into a double integral over the region enclosed by curve \(C\). The integral simplifies to:

\[ -\frac{3}{2} \pi a^3, \]

considering the specified vector field components.