Question

Valuate the line integral, where c is the given curve. ∫c zdx + xdy + ydz

Answer 1

$\displaystyle\int_{\mathcal C}z\,\mathrm dx+x\,\mathrm dy+y\,\mathrm dz=\int_{\mathcal C}\langle z,x,y\rangle\cdot\mathrm d\langle x,y,z\rangle$
$\displaystyle=\int_{t=0}^{t=1}\langle t^2,t^2,t^3\rangle\cdot\mathrm d\langle t^2,t^3,t^2\rangle$
$=\displaystyle\int_0^1\langle t^2,t^2,t^3\rangle\cdot\langle2t,3t^2,2t\rangle\,\mathrm dt$
$=\displaystyle\int_0^15t^4+2t^3\,\mathrm dt=\frac32$