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zvonat [6]
3 years ago
13

Evaluate the integral Integral from left parenthesis 2 comma 1 comma 2 right parenthesis to left parenthesis 6 comma 7 comma neg

ative 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.
Mathematics
1 answer:
White raven [17]3 years ago
8 0

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

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