Question

Find the length of the curve

x=1−t,y=t−5,z=2t−2,

  for 1≤t≤3

Answer 1

To find length of curve use:

`\int\limits^a_b r'(t) \, dt`
where r(t) is vector equation:

`r(t) = (1,-5,-2) + (-1,1,2)t`

`r'(t) = (-1,1,2)`
Magnitude of vector r'(t) is then:

`√((-1)^2 +1^2 +2^2) =√(6)`
Now evaluate integral

`\int\limits^3_1 {√(6)} \, dt = √(6) (3-1) = 2 √(6)`