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MathSofiya
just making sure I understand it.... \[S=\int_{a}^{b} 2\pi y \sqrt{1+{(\frac{dy}{dx})}^2} dx\] \[S=\int_{c}^{d} 2\pi y \sqrt{1+{(\frac{dx}{dy})}^2} dy\] \[S=\int_{a}^{b} 2\pi x \sqrt{1+{(\frac{dy}{dx})}^2} dx\] One is for revolution about the x axis...the other for revolution about the y- axis...and then we have one more equation....why?
Area of a Surface of Revolution
let me make some drawing
we are taking circumference of each plate , they have height(radius) of 'y' 2 Pi r= 2 Pi y
what if we're rotating about the y axis...would it still be 2 pi y
it would be much more complicated
first ,you gotta define function in term of x
it is too complicated unlike finding volume , we just use x axis
|dw:1341859119434:dw|
\[S=\int 2\pi x ds\]
but the book still has (dy/dx)^2
@helder_edwin please help
give a second to check my books it's been a long time
Everyone abandoned me :'(
I didn't get your ping... weird I'll have to ask administration about that... ok, what do we ave here, let me read...
so you want to know why we have 4 formulas, right? my answer is that there are really only two, but each one can be seen from two different perspectives...
first consider the arc length formula:\[ds=\sqrt{1+[f'(x)]^2}dx\]now this is the formula for arc length taken from the perspective of y being a funcion of x but arc length is the same regardless of whether you look at the function as f(x) or g(y) since the arc itself will still have the same length. so we can also write\[ds=\sqrt{1+[g'(y)]^2}dy\] and as long as we are talking about the same curve they should be equal, since the arc can only have one length. make sense so far?
now for a revolution, the formula is\[A=\int2\pi yds\]or\[A=\int2\pi xds\]depending on which axis we are going around but as I just explained above, ds (the arc length differential) can always be written two ways depending on whether we consider y a function of x or vice-versa, so each of these formulas is potentially two depending on how we look at our ds
can you read spanish?
here is a nice full explanation if you care to dig deeper http://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx
I will , thanks @TuringTest .
Sofiya, try to break down each integral like this: First, look at the end part, the variable you're integrating with respect to. Then, look at the limits, so you say to yourself, "Okay, travelling along x from a to b." or something like that. Then, look at the function inside and try to break that down, and what that means at each particular x. For these particular integrals, the insides have 2 basic parts. The first part has the form 2pi*something, where that something is the radius. The second part is the arclength formula. So it's calculating the arclength, and then it's multiplying that in a circle.
sorry i wanted to send you something but it's in spanish. but i goes much in the same way as what @TuringTest did
oh ok....thanks everyone Thanks @SmoothMath !!!
@helder_edwin me lo mandas por favor? quiero aprender mas la terminologia en espanol
@TuringTest recibiste el pdf?
no, donde lo pusiste? podrias mandarme un ("link"?) o url ? ya puesto que estoy tu "fan" es posible mandar mensajes privadas