Find the line integral of f(x,y,z) = x + y + z over the straight-line segment from (5,4,3) to (4,1,1).

Question

Find the line integral of ​f(x,y,z) = x + y + z over the​ straight-line segment from ​(5​,4​,3​) to ​(4​,1​,1​).

Vocaloid · Answer

Please do not comment unless you are actually familiar with the subject and are making a genuine attempt to help.

sillybilly123 · Answer

$f(\vec x) = x + y + z$ from (5,4,3) to (4,1,1) $\vec r = <5 , 4, 3> + t <-1, -3, - 2>$ $= <5 - t , 4 - 3t, 3- 2t>, ~ ~ ~ ~ 0 \le t \le 1$ $S = \int\limits_0^1 ~ ds ~~~~ x + y + z $ $ = \int\limits_0^1 ~ \sqrt{\dot x^2 + \dot y^2 + \dot z^2} ~ dt ~~~~ x + y + z $ $ = \int\limits_0^1 ~ \sqrt{(-1)^2 + (-3)^2 + (-2)^2} ~ dt ~~~~ (5 - t) + (4 - 3t) + (3- 2t) $ $ = 6\sqrt{14}\int\limits_0^1 ~ ~ dt ~~~~ 2 - t$ $ = 6\sqrt{14} ~~~ \left[2t - \frac{1}{2}t^2 \right]_0^1 = 9 \sqrt{14}$ Hopefully 🥴