Question

The value of the integral $$ \oint_C \left[ (x^3 + xy) \, dx + (x^2 - y^3) \, dy \right] $$ where $C$ is the square formed by the lines $x = \pm 1$, $y = \pm 1$, is: 

• A. 1
• B. \( \frac{1}{8} \)
• C. \( \frac{1}{4\sqrt{2}} \)
• D. 0

Hint:

Using Green’s Theorem simplifies evaluating line integrals by converting them to double integrals over the enclosed region.

Answer 1

The Correct Option is D

Green's Theorem converts this closed-path line integral into a double integral over the enclosed area. The result of this double integral is zero.