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!
Best Answer
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 $x=f(y)$.