Suppose $\overrightarrow F(x,y,z) = -y\overrightarrow i + x\overrightarrow j + 0.5\overrightarrow k$ and $C$ is the helix given by $x(t) = 5 cos (t), y(t)=5sin(t),z(t)=t/5$ for $0 \leq t\leq6\pi$. Find the line integral.
So, here is what I did. $\int_C\overrightarrow F \bullet d\overrightarrow r) = \int_0^{6\pi}(-5\sin (t)+5\cos (t)+0.5)(\sqrt {25\sin^2 t+25\cos^2 t+1/25})$
I have this gut feeling that I messed up the $r't$ part of $\int_a^b f(r(t))|r't|dt $ but I can't quite figure the reason out.
You have $\overrightarrow F(x,y,z) = -y\overrightarrow i + x\overrightarrow j + 0.5\overrightarrow k$ and $C$ is the helix given by $x(t) = 5 cos (t), y(t)=5sin(t),z(t)=t/5$ for $0 \leq t\leq6\pi$. Take
$\int_C\overrightarrow F \bullet d\overrightarrow r) = \int_0^{6\pi}(-5\sin (t), 5\cos (t), 0.5). (-5 sin(t), 5\cos(t), 1/5)dt$ = $\int_0^{6\pi} (25 + 1/10 )dt$
$=(25 + 1/10 ) 6\pi.$
NB: The formula you have written for the line integral is NOT correct. There should not be any absolute value sign.