Line, Surface and Volume Integrals

Evaluate the line integral \(\int_C[(x^2+y^2)\vec i-2xy\vec j]\cdot d\vec r\), where C is the parabola \(x^2=y\) from (0,0) to (1,1).

Given the vector field \(\vec F=\sin y\vec i+x(1+\cos y)\vec j\), evaluate the line integral over the circular path given by \(x^2+y^2=a^2,\ z=0\).

A vector field is given by \(\vec F=\sin y\vec i+x(1+\cos y)\vec j\). Evaluate the line integral over the circular path given by \(x^2+y^2=a^2,\ z=0\).