Question

integra la funcion f(x,y,z)=3x+y+4z^3 sobre la trayectoria x=3t,y=-t,z=-1/3t para 5<=t<=7

Answer 1

$\int_Cfds$

como sabemos 
$ds=\sqrt{x_t^2+y_t^2+z_t^2} dt\\ ds=\sqrt{9+1+1/9} \,\,dt\\ ds=\sqrt{91}/3 \,\,dt$

$f(x,y,z)=F(t)=3(3t)-t+4(-1/3 t)^3\\ F(t)=8t-4t^3/27$

entonces
$\int\limits_5^7 (8t-4t^3/27) \sqrt{91}/3 \,\,dt = \sqrt{91}/3 \int\limits_5^7 8t-4t^3/27 dt=272\sqrt{91}/27$