Find the line integral of $f ( x , y , z ) = x + y + z$ over the straightline segment from $( 1,2,3 )$ to $( 0 , - 1,1 )$ .
Okay. What we want to do is we want to find the line integral. Um uh f of X y z equal to X plus y Percy over the straight line. Um, segment from one comma, two comma. Three 20 Negative. 11 Okay, so what we need to do is first of all, um, we need t determine, X. Why N c? And so, um, if I go from ex of one 20 um, that is going to be a T. If I go from why? From two to negative one. I've gone, um, three t. And if I go from three toe one in the Z, I actually have gone to t. Okay, Um and so now, um, we need to define a, um, or tea is equal to t. I play three t j plus two tea. Okay, um, and T is just gonna go from 0 to 1, okay. And so now I need to take, um I need to find V A t, which is equal to the derivative of our and so this is gonna be I plus three j plus two K. And so the magnitude of the is equal to the square root of one squared plus three squared plus two squared which is going to give me the square root of 14. Um and so D s is equal to the square root of 14 DT. And so, um, the integral become 0 to 1. Uh um t x plus y posie, which is t plus three t plus two t times the square root of 14 TT. So this is gonna be equal. Thio, we have 62. This is gonna be six times the square root of 14 times the integral from zero toe one uh, T T t. Which is gonna give me six times the square root of 14 times one have t squared from 0 to 1, and so this will be three square root of 14.