Evaluate the integral integral from (5 comma 2 comma 4 )to (7 comma 9 comma negative 3 )y dx plus x dy plus 3 dz by finding parametric equations for the line segment from (5 ,2,4) to (7 ,9,negative 3) and evaluating the line integral of fequals yiplusxjplus3k along the segment. since f is conservative, the integral is independent of the path.

Parameterize the line segment (call it ) by

with . Then

and the line integral is

Alternatively, if we can show that is conservative, then we can apply the fundamental theorem of calculus. We need to find such that , which requires

Integrating both sides of the first equation with respect to gives

Differentiating both sides wrt gives

Differentiating wrt gives

So we have

and is conservative. By the FTC, we find

5x + 3y = - 4

step-by-step explanation:

the equation of a line in standard form is

ax + by = c ( a is a positive integer and b, c are integers )

express the line first of all in slope- intercept form

y = mx + c ( m is the slope and c the y-intercept )

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

with (x₁, y₁ ) = (7, - 3) and (x₂, y₂ ) = (4, - 8)

m = = -

y = - x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (4, - 8 ), then

- 8 = - + c ⇒ c = -

y = - x - ← in slope-intercept form

multiply all terms by 3 to eliminate the fractions

3y = - 5x - 4 ( add 5x to both sides )

5x + 3y = - 4 ← in standard form