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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!

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    You can probably just use Google Books. It would make your life a bit easier, along with ours.2012-02-27

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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 $x=f(y)$.

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    Wait a minute, why does your last integral formula of 2pi$y$sqrt(...) instead of 2pi$x$sqrt(...)2015-04-10
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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.