How Do You Calculate the Line Integral of a Polygonal Path?

[jackalope1234]

Homework Statement

Evaluate Integral y^2 dx + (xy - x^2) dy over the given path C (0,0) to (2,4)
the polygonal path (0,0), (2,0), (2,4) (All one question)

Homework Equations

integral of h (dot product) dr over C

The Attempt at a Solution

I realize I have to parametrize the line segments and use the formula on them individually but whenever I try to parametrize the line segments and then sub what they equal into the equation I end up getting a 0.

for instance (assuming 3 separate segments)

C1. since < 2 , 0 >
x = 0 + 2t, y = 0 + 0t

upon subbing this into the equation I am left with
integral of 0.

I'm guessing I'm parametrizing them wrong but I don't know how.

 

[LCKurtz]

Homework Statement

Evaluate Integral y^2 dx + (xy - x^2) dy over the given path C (0,0) to (2,1)
the polygonal path (0,0), (2,0), (2,4) (All one question)

Homework Equations

integral of h (dot product) dr over C

The Attempt at a Solution

I realize I have to parametrize the line segments and use the formula on them individually but whenever I try to parametrize the line segments and then sub what they equal into the equation I end up getting a 0.

for instance (assuming 3 separate segments)

C1. since < 2 , 0 >
x = 0 + 2t, y = 0 + 0t

upon subbing this into the equation I am left with
integral of 0.

I'm guessing I'm parametrizing them wrong but I don't know how.

For one thing, you have a typo. The path is supposed to go from (0,0) to (2,1) but your polygonal path ends at (2,4), so something is amiss.

But to address your concern, it is OK for line integrals to come out any number, including 0. And on that segment both y and dy are 0 making your integrand 0 as you have correctly calculated. Nothing wrong so far.

 

[HallsofIvy]

From (0, 0) to (2, 0), let x= t, y= 0. Then dx= dt, dy= 0, while your integrand becomes
$y^2= 0$, so the integral is
[tex]\int_0^2 0dt+ (xy- x^2)0= 0[/itex]

From (2, 0) to (2, 4), let x= 2, y= t. Then dx= 0, dy= dt, while your integrand is $xy- x^2= 2t- 4$ so the integral is
$$\int_0^4 y^2(0)+ (2t- 4)dt= \int_0^4 (2t- 4)dt$$