## MathSofiya 3 years ago 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?

1. Libniz

what the devil is this?

2. MathSofiya

Area of a Surface of Revolution

3. Libniz

surface area ?

4. MathSofiya

yes

5. Libniz

let me make some drawing

6. MathSofiya

@TuringTest

7. Libniz

|dw:1341858624540:dw|

8. Libniz

we are taking circumference of each plate , they have height(radius) of 'y' 2 Pi r= 2 Pi y

9. MathSofiya

what if we're rotating about the y axis...would it still be 2 pi y

10. Libniz

no

11. Libniz

it would be much more complicated

12. MathSofiya

2 pi x

13. Libniz

not that simple

14. Libniz

first ,you gotta define function in term of x

15. MathSofiya

ok

16. MathSofiya

as in x= .....y....

17. MathSofiya

or x=g(y)

18. Libniz

it is too complicated unlike finding volume , we just use x axis

19. MathSofiya

ok

20. MathSofiya

|dw:1341859119434:dw|

21. MathSofiya

$S=\int 2\pi x ds$

22. MathSofiya

but the book still has (dy/dx)^2

23. MathSofiya

24. Libniz

it is 3 dimensional

25. helder_edwin

give a second to check my books it's been a long time

26. MathSofiya

ok

27. MathSofiya

Everyone abandoned me :'(

28. helder_edwin

no

29. TuringTest

30. TuringTest

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...

31. MathSofiya

ok

32. TuringTest

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?

33. MathSofiya

Yes

34. TuringTest

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

35. helder_edwin

36. TuringTest

yo si

37. MathSofiya

sorry I can't

38. SmoothMath

lol. That guy.

39. TuringTest

here is a nice full explanation if you care to dig deeper http://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx

40. MathSofiya

I will , thanks @TuringTest .

41. TuringTest

welcome :)

42. SmoothMath

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.

43. helder_edwin

sorry i wanted to send you something but it's in spanish. but i goes much in the same way as what @TuringTest did

44. MathSofiya

oh ok....thanks everyone Thanks @SmoothMath !!!

45. SmoothMath

My pleasure =D

46. TuringTest

@helder_edwin me lo mandas por favor? quiero aprender mas la terminologia en espanol

47. helder_edwin

claro!

48. helder_edwin

@TuringTest recibiste el pdf?

49. TuringTest

no, donde lo pusiste? podrias mandarme un ("link"?) o url ? ya puesto que estoy tu "fan" es posible mandar mensajes privadas