Surface area formula I'm kind of confused about the explanation of the surface area formula in my text book 
The text gave us $$\int_{a}^{b}2\pi f(x) \sqrt{1+[f'(x)]^2}dx$$ after that the formula is getting like 
$$\int_{a}^{b}2\pi y \sqrt{1+\left(\frac{dy}{dx}\right)^2 }dx$$
also it said 
$$\int_{a}^{b}2\pi y \sqrt{1+ \left(\frac{dx}{dy}\right)^2 }dy$$
They used $y$ instead of $f(x)$. It confused me.
How can I figure out between when I plug in some $y$ function at $y$ and when I change any thing (just use $y$) in the formula?
Kind of difficult to explain though, I hope you can understand my question. 
And if you have any idea, please post it. Thank you! 
 A: It is not that hard.
The first formula is given for $$y = f(x)$$
So, you just choose a function and replace both $f(x)$ and $f'(x)$.
The second formula is simply using the $y$ notation instead of $f(x)$. So there you have again $$y = f(x)$$
and they write $$f'(x) = \frac{dy}{dx}$$
You can interpret the last formula as follows: suppose the $y$ axis is the independent variable and $x$ is the dependent variable, then
$$\int\limits_a^b {2\pi y\sqrt {1 + {{\left( {\frac{{dx}}{{dy}}} \right)}^2}} dy} $$
gives the surface of revolution around the $y$ axis - here we have $x=f(y)$.
A: What you are looking at is the formula for the area of the surface generated by revolving the curve y = f(x) about the x-axis.  There are four parts to the formula a = lower bound of integration, b = upper bound of integration (the interval on the x-axis which the function is being defined), y = f(x) (the actual function being rotated that is in terms of x, and dy/dx (the derivative of the formula being rotated).  
The formula for rotations about the y-axis will use the dx/dy and x=g(y) (a function defined in terms of y).
I think you are getting the two of them mixed up.
