Question

If \[ I=\int_C x^2y\,dx-xy^2\,dy, \] where \(C\) is the circle \[ x^2+y^2=16, \] then \(2I=\)

• A. \(-128\pi\)
• B. \(-256\pi\)
• C. \(-512\pi\)
• D. \(0\)

Hint:

For a closed curve, \[ \boxed{ \oint_C M\,dx+N\,dy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right)dA } \] Use polar coordinates whenever the region is a circle.

Answer 1

The Correct Option is B