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

The given integral is a line integral around a closed path. We apply Green’s Theorem to convert the line integral into a double integral over the area enclosed by the curve. After calculating the double integral, we find that the result is zero.