Question

Hi, I'm working on line integrals. I have a specific problem, but I keep getting stuck when I try to find the parameters (in t). For example: F= r(t)=?

Answer 1

again the question isn't specific please make it!

Answer 2

you mean, u have a vector valued function, and the question asks to find the distance between 2 coordinate points while your upper and lower boundary of the integral has to be in terms of t ?

simple: get the value of t's by plugging the initial coordinate (either x, or y) in the respective component of the vector function then find for t, do same for final value.

example:

r(t)=sin(t)i-cos(t)j+tk
find distance from (0,0,0) to (1,1,1)

as you see, the initial value and final value of t is not given, you have to find it from the initial and final coordinates of the motion.

sin(t)=0, therefore, t=0 <- Initial t
sin(t)=1, therefore, t=pi/2 <- final t

you could use and point of the coordinates and plug in the respective component of vector function and you will get the same initial and final t.

\[\int_{0}^{\pi/2} |v(t)| dt\]