Question

Evaluate the integral f*dr where f(x,y)=<-y,x^2> and c consists of te arc of the parabola y=2x^2 from (-1,2) to (1,2)

Answer 1

$y=2x^2\implies\mathbf r(x)=\langle x,2x^2\rangle$

$\displaystyle\int_C\mathbf F\cdot\mathrm d\mathbf r=\int_{t=-1}^{t=1}\langle-2x^2,x^2\rangle\cdot\langle1,4x\rangle\,\mathrm dx$
$=\displaystyle\int_{-1}^1(8x^3-2x^2)\,\mathrm dx$
$=-\dfrac43$