# Integrals 3

1. Jan 25, 2012

### bugatti79

1. The problem statement, all variables and given/known data

Evaluate this integral along the line segment from (0,1) to (pi, -1) by parameterising this segment

2. Relevant equations

$\int y sin x dx - cos x dy$

3. The attempt at a solution

How would I parameterise this line segment?

2. Jan 25, 2012

### SammyS

Staff Emeritus
Start by writing an equation for the line passing through those two points.

3. Jan 25, 2012

### bugatti79

$y= \frac{-2x}{\pi} +1$. How do I know whether to let x=t or not?

4. Jan 25, 2012

### tiny-tim

hi bugatti79!
you can let t = x, or t = y, or t = any function of x and y so long as it's strictly increasing (or decreasing) along the line

5. Jan 25, 2012

### SammyS

Staff Emeritus
You may want t = 0 to correspond to x = -1 (the left end of the interval) and t = 1 to correspond to x = 1 (the right end of the interval).

That's a fairly common practice.

6. Jan 26, 2012

### bugatti79

Very good, thats a good tip. Will keep that in mind.

I dont follow this to be honest. I am not sure what the line is. Is it an arc from (0,1) to (pi, -1)?

Does this mean the integral will evaluate to 0? I think that is only for closed simple curves and we are dealing with an open line hence it will be none 0..right?

What is the significant of whether it was a segment or not?

7. Jan 26, 2012

### SammyS

Staff Emeritus
The problem states LINE SEGMENT from (0,1) to (π, -1). That means the portion of the (straight) line passing through the points which is between (0,1) and (π, -1) including the endpoints.

The simplest parametrization is to let x(t) = t, and $\displaystyle y(t)=\frac{-2x(t)}{\pi} +1=\frac{-2t}{\pi} +1\,,$ where t goes from 0 to π.

A parametrization I like is for t to go from 0 to 1. Then x(t) = π t, and $\displaystyle y(t)=\frac{-2\pi t}{\pi} +1=-2t+1\,.$

8. Jan 26, 2012

### bugatti79

Ok. I will work out both. Trying the first I get

$\displaystyle \int y sin x dx - cos x dy = \int (\frac{-2 t}{\pi}+1) sin t dt+ \frac{2}{\pi} cos t dt=0$......