Question

The value of the integral \( \int_C (2xy - x^2) \, dx + (x^2 + y^2) \, dy \) where \( C \) is the boundary of the region enclosed by \( y = x^2 \) and \( y^2 = x \), described in the positive sense, is

• A. \(-2\)
• B. \(0\)
• C. \(-1\)
• D. \(2\)

Hint:

When given a line integral over a closed curve, check if Green’s theorem can be used to convert to a simpler double integral.

Answer 1

The Correct Option is B

Use Green’s Theorem: \[ \oint_C M\,dx + N\,dy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx\,dy \] Given: \[ M = 2xy - x^2,\quad N = x^2 + y^2 \] Then: \[ \frac{\partial N}{\partial x} = 2x,\quad \frac{\partial M}{\partial y} = 2x \Rightarrow \text{Integrand} = 2x - 2x = 0 \] So the double integral over region \( R \) is 0.