Quick question about taking a derivative

[tnutty]

So say for a line integral, the curve C is given by y = sqrt(x), from point (1,1) to (4,2);

In my integral I have some integrand* dy.

Say I wanted to change the dy to dx.

From what's given :

y = sqrt(x);

dy = 1/(2sqrt(x) ) dx;

I could just substitute that instead for dy.

But what's the difference if I do this :

y = sqrt(x)
y^2 = x

2y dy = dx

dy = dx/2y

So how is the former different from the latter. I mean I see that y is integrated for the
second one, but what does it represent? Can you explain me the difference between the
two, does not have to be geometrically, but will be appreciated.

 

[grief]

Since y=sqrt(x), 1/(2sqrt(x) ) dx =dx/2y. However, the first form is the one you want to use, because it's strictly in terms of x.

 

[LCKurtz]

The parabola can be represented by either y = sqrt(x) and x = y² on that interval. Which you use might be determined by what the rest of the integrand is. Or maybe you have an aversion to square roots in integrals. Use whichever one looks easiest in your problem.