Assorted line integrals Evaluate the line integral using the given curve C. + y) dx + (x – y) dy + x dz; C is the line segment from(1, 2, 4) to (3, 8, 13).

The given line integral ∫Cx+ydx+x-ydy+xdz, where C is the line segment from 1, 2, 4 to 3, 8, 13. We…

Answered: + y) dx + (x – y) dy + x dz; C is the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.1: Rectangular Coordinate Systems

Problem 35E

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Question

Assorted line integrals Evaluate the line integral using the given curve C.

+ y) dx + (x – y) dy + x dz; C is the line segment from
(1, 2, 4) to (3, 8, 13).

Expert Solution

Step 1

The given line integral $\int_{C} (x + y) d x + (x - y) d y + x d z$ , where $C$ is the line segment from $(1, 2, 4)$ to $(3, 8, 13)$ .

We have to find the value of the line integral.

Step 2

The given line integral $\int_{C} (x + y) d x + (x - y) d y + x d z$ , where $C$ is the line segment from $(1, 2, 4)$ to $(3, 8, 13)$ .

The parametric equation of the line segment,

$\begin{array}{rcl} x & = & x_{1} + t (x_{2} - x_{1}) \\ = & 1 + t (3 - 1) \\ x & = & 1 + 2 t \end{array}$

$\begin{array}{rcl} y & = & y_{1} + t (y_{2} - y_{1}) \\ = & 2 + t (8 - 2) \\ y & = & 2 + 6 t \end{array}$

$\begin{array}{rcl} z & = & z_{1} + t (z_{2} - z_{1}) \\ = & 4 + t (13 - 4) \\ z & = & 4 + 9 t \end{array}$

We get $x (t) = 1 + 2 t, y (t) = 2 + 6 t,$ and $z (t) = 4 + 9 t$ .

Differentiate with respect to $t$ ,

$x' = 2, y' = 6,$ and $z' = 9$ .

For $t = 0$ , we get $(x, y, z) = (1, 2, 4)$

For $t = 0$ , we get $(x, y, z) = (3, 8, 13)$ .

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+ y) dx + (x – y) dy + x dz; C is the line segment from (1, 2, 4) to (3, 8, 13).