Evaluating Line Integrals Along a Curve

[bfr]

If a question says something like: "evaluate \int(x*z*y)dx - (x-y)dy + (x^3)dz from (1,0,0,) to (1,0,2pi) along the curve (x,y,z)=(cos(t),sin(t),t)" or something like that, this is just basically splitting up a line integral? In my example, it would be the same as: \intcos(t)*t*sin(t)) * (-sin(t)) dt - \int(cos(t)-sin(t))*cos(t) dt ... etc. , which is just: \int<(cos(t)*t*y),-(cos(t)-y),(cos(t)^3)> dot <-sin(t),cos(t),1> dt from t=0 to t=1 ("dot" represents a dot product).

 

[Unco]

If a question says something like: "evaluate \int(x*z*y)dx - (x-y)dy + (x^3)dz from (1,0,0,) to (1,0,2pi) along the curve (x,y,z)=(cos(t),sin(t),t)" or something like that, this is just basically splitting up a line integral? In my example, it would be the same as: \intcos(t)*t*sin(t)) * (-sin(t)) dt - \int(cos(t)-sin(t))*cos(t) dt ... etc. , which is just: \int<(cos(t)*t*y),-(cos(t)-y),(cos(t)^3)> dot <-sin(t),cos(t),1> dt from t=0 to t=1 ("dot" represents a dot product).

Pretty much, though the limits would be from t=0 to t=2pi, and it's the second form of the integral (i.e. the expanded dot product) that is useful to calculate (by evaluating the definite integral with respect to t).

 

[bfr]

OK, thanks.

And, er, yeah, I meant from t=0 to t=2pi.