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The paths of integration for Exercises 15 and 16

Integrate $f ( x , y , z ) = x + \sqrt { y } - z ^ { 2 }$ over the path from $( 0,0,0 )$ to $( 1,1,1 )$ (see accompanying figure) given by

$$

\begin{array} { l l } { C _ { 1 } : } & { \mathbf { r } ( t ) = t \mathbf { k } , \quad 0 \leq t \leq 1 } \\ { C _ { 2 } : } & { \mathbf { r } ( t ) = t \mathbf { j } + \mathbf { k } , \quad 0 \leq t \leq 1 } \\ { C _ { 3 } : } & { \mathbf { r } ( t ) = t \mathbf { i } + \mathbf { j } + \mathbf { k } , \quad 0 \leq t \leq 1 } \end{array}

$$

$-\frac{1}{6}$

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No, I'm looking for the Pathans. A girl of F over a path. See which is connected as C one C two. Let's see three. Come still, Buy the property of Pathan's a girl. This equals two that sum it up. So, case from 1 to 3, we have ck fds. So that's how we're gonna find this Pathan to go. Okay, so no, we have seen once it's UNC three three paths respectively. So especially divide and conquer of we're gonna find this Pathan to grow on those sup sup path or accept them for respecting London. Adam up to get off. Original question to see one. Let's see, we have our one or two or three. The stands for this stand for the path of C one sea to see through the Parametric function for singer through C three, respectively. So for our one, let's see, it's Tatum's case of this is 001 zeros on T for or two, huh? It's zero C one for three is C 11 and oversee his perimeter. Rosa is always from 0 to 1. Okay, so no way establishing. We're gonna establish these passenger girl over three paths respectively. so recall the method to transformed a path and a girl into a normal into girl with respect to the parameter t. So how do we do that? The first is gonna substitute the X y and Z in our inter grant by t. And the 2nd 1 is two multiplied by the norm of the derivative or so, which is a normal another factor, um, the better of derivative. So what's that is done? We can transform it to they lower bound to upper bound and d t. So this place we have works, Let's see. So, for 1st 1 we have the function is like this Sawyer time stay, um, nor malicious one. So the answer is naked 1/3 for the 2nd 1 The function will be squirrel team US one and a woman is also one. And the answer for that iss also swear up one over three. Negative. For the last part, the int'l grants will be Tatum's one. So taste the function one. It's the norm of the true motive. And this is when Hef So now Adam up we have half on C. D s equals two won over three minus one over three times to pores. What half? So the answer is nothing. 1 60

University of Illinois at Urbana-Champaign