Hi, I've just started working on line integrals and I don't understand one of the examples in my book.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\int\limits_C {y^2 dx + xdy}

[/tex]

Where C is the arc of the parabola x = 4 - y^2 from (-5,-3) to (0,2). The book proceeds by suggesting that y is taken as the parameter so that the arc C is represented by x = 4 - y^2, y = y and -3 <= y <= 2.

Then dx = -2ydy so that [tex]\int\limits_C {y^2 dx + xdy} = \int\limits_{ - 3}^2 {y^2 \left( { - 2y} \right)dy} + \left( {4 - y^2 } \right)dy = ... = 40\frac{5}{6}[/tex].

So one of the variables (in this case y) has been taken as the parameter. I am wondering why that decision was made. Why note take x as the parameter? Is it because taking y as the parameter leads to a simpler calculation.

I am also having having trouble understanding the difference between a line integral wrt to an arc length and a line integral wrt x or y. I have the following formula for the line integral with respect to arc length.

[tex]

\int\limits_C {f\left( {x,y} \right)} ds = \int\limits_a^b {f\left( {x\left( t \right),y\left( t \right)} \right)} \sqrt {\left( {\frac{{dx}}{{dt}}} \right)^2 + \left( {\frac{{dy}}{{dt}}} \right)^2 } dt

[/tex]...equation (1)

The formula for the integral with respect to the y is:

[tex]

\int\limits_C {f\left( {x,y} \right)} dy = \int\limits_a^b f \left( {x\left( t \right),y\left( t \right)} \right)y'\left( t \right)dt

[/tex]...equation (2)

My confusion basically comes down to not understanding the difference between equations (1) and (2). In equation (1) is the integration along some curve? In equation (2) is the integration along the y-axis? If so then how come to example above, y is chosen as the parameter rather than using t as the parameter as equation (2) suggests it should be done.

I know that in this case using y, or t or pretty much any other variable as the parameter makes no difference to the value of the parameter. It just seems strange that the example is not consistent with equation (2).

The above is probably quite confusing by any help would be great thanks.

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# Homework Help: Line integral examples in book

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