Evaluate the line integral $\int_C zdx+xdy+ydz $

Question

Calculate $\int_C zdx+xdy+ydz $ where $C= \{(x,y,z) \;| \;x=t, y=t^2, z=t^3, 0 \leq t \leq 1 \} $

So

$$x=t $$ $$dx=1 $$ $$y=t^2 $$ $$dy=2t $$ $$z=t^3 $$

$$dz=3t^2 $$

$$\int_{0}^{1} [t^3(1)+t(2t)+t^2(3t^2)]dt $$

$$\int_{0}^{1} (t^3+2t^2+3t^4)dt $$

$$[\frac{t^4}{4}+\frac{2t^3}{3}+\frac{3t^5}{5}]_{0}^{1} $$

$$=\frac{91}{60} $$

Answer 1

This is correct, although you should write $dx=1dt$, $dy=2tdt$ and $dz=3t^2dt$, to be precise.