Answer
$\frac{722}{15}$
Work Step by Step
$\int_{c} y dx+ z dy +x dz $; C: $ x= \sqrt t, y=t, z=t^{2}$ Line integral, $1\leq t\leq 4$
$\displaystyle\int _{1}^{4}\left( t\times\frac{1}{2\sqrt t}+t^2+\sqrt t\times2t \right ) dt $
$=\displaystyle\int _{1}^{4}\left(\frac{1}{2}\times\sqrt t+t^2+2t\sqrt t\right) dt$
$=\left(\frac{t^{3/2}}{2(3/2)} +\frac{t^3}{3}+\frac{2t^{5/2}}{5/2}\right)\bigg |_1^4$
$=\frac{8}{3}+\frac{64}{3}+\frac{4(32)}{5}-\frac{1}{3}-\frac{1}{3}-\frac{4}{5}$
$=\frac{70}{3} + \frac{124}{5}$
$=\frac{722}{15}$
