Question

Can anybody help me with this problem regarding Line Integrals (Calc 3)?

Compute ∫F*dr, given the counterclockwise unit circle C : cos((pi)t), sin((pi)t), t∈[0,2] and the vector field F(x,y) = (y²,-x²)

Answer 1

The line integral is

$\displaystyle \oint_C\mathbf F(x,y)\cdot\mathrm d\mathbf r = \int_0^2 \mathbf F(x(t),y(t))\cdot\frac{\mathrm d\mathbf r}{\mathrm dt}\,\mathrm dt \\\displaystyle= \int_0^2 (\sin^2(\pi t),-\cos^2(\pi t))\cdot(-\pi\sin(\pi t),\pi\cos(\pi t))\,\mathrm dt \\\displaystyle=-\pi\int_0^2(\sin^3(\pi t)+\cos^3(\pi t))\,\mathrm dt = \boxed{0}$