Question

Evaluate the integral Integral from left parenthesis 2 comma 1 comma 2 right parenthesis to left parenthesis 6 comma 7 comma negative 5 right parenthesis y dx plus x dy plus 7 dz by finding parametric equations for the line segment from ​(2​,1​,2​) to ​(6​,7​,negative 5​) and evaluating the line integral of Fequalsyiplusxjplus7k along the segment. Since F is​ conservative, the integral is independent of the path.

Answer 1

ANSWER

-9

EXPLANATION

We want to evaluate the line integral:

$\int_{(2,1,2)}^{(6,7,-5)}ydx+xdy+7dz$

The parametric equations for the line segment from (2,1,2) to (6,7,-5) is

$x = 2 + 4t$

$y= 1 + 6t$

$z= 2 - 7t$

This implies that:

$dx = 4dt$

$dy = 6dt$

$dz = - 7dt$

Our integral now becomes:

$\int_{0}^{1} (1 + 6t)(4)+(2 + 4t)(6) + 7( - 7)dt$

We simplify to get:

$\int_{0}^{1} 48t - 33dt$

This integral evaluates to:

$\int_{0}^{1} 48t - 33dt = - 9$