# Finding limits of line integral

• boneill3
In summary, the conversation is about finding the line integral along a segment from (0,0) to (\pi,-1) using the formula \int_{a}^{b}[f(x(t),y(t))x'(t) + g(x(t),y(t))y'(t)]dt with a given parameterization. The problem arises when trying to change the limits from the original variables (x,y) to the parameter t, and the individual is seeking help in finding the correct limits for t. They also mention that the question states the integral is independent of path and they are trying to verify this by integrating along the initial line segment.

## Homework Statement

Integrate along the line segment from (0,0) to $(\pi,-1)$
The integral

$\int_{(0,1)}^{(\pi,-1)} [y sin(x) dx - (cos(x))]dy$

## The Attempt at a Solution

I have used the parameterization of $x=\pi t$ and $y= 1-2t$
To get the integral:
$\int_{(0,1)}^{(\pi,-1)} [1-2t sin(\pi t) -(cos(\pi t))]dt$

But now because it is an integral of variable t I need to change the limits .

I'm not sure if I just have to put the limits of t just from 0 to $\pi$

I suppose I'm having trouble with getting from the limit of 2 variables (x,y) to a limit of one variable t

Thanks

If x=pi*t and y=1-2t, then if you put t=0 then x=0 and y=1, right? If you put t=1 then x=pi and y=(-1), also right? As you came up with that fine parametrization what's the problem with finding limits for t?

I will need to go back and study more about parametrization.

When calculating this line integral

$\int_{(0,1)}^{(\pi,-1)} [y sin(x) dx - (cos(x))]dy$

I'm using the formula
$\int_{a}^{b}[f(x(t),y(t))x'(t) + g(x(t),y(t))y'(t)]dt$

with parameterization

I have $x = \pi t$
$y = 1-2t$
so
$x' = \pi$
and
$y' = -2$

plugging into the integral I get

$\int_{(0)}^{(1)} [1-2y sin(\pi t) \pi - (cos(\pi t))-2]$
$= -1$

The question states that the integral is independant of path.

So if I integrate along the initial line segment $(0,1)$to $(0,\pi)$
I should be able to plug in the values f($(-1,\pi)$)-f($(0,1)$)
And it should equal my original integral vaue of -1.

However I get 0

Could someone please check what I've done I show me where I am going wrong ?