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Find the area of the surface of the curve obtained by rotating the

  1. Feb 10, 2013 #1
    "Find the area of the surface of the curve obtained by rotating the.."

    1. Find the area of the surface obtained by rotating the curve y= 1+5x^2

    from x=0 to x=5 about the y-axis.


    2. I thought to find surface area, we would need to use this formula:

    SA= ∫2[itex]\pi[/itex](f(x))√(1 + (f'(x))2)]dx


    3. So far I have:

    ∫0 to 5 of 2[itex]\pi[/itex][(1 + 5x2)(√(1+100x2))]dx


    I first tried to do a u substitution for what's in the square root (1 + 100x2), but then du only equals
    200x... And I'm not able to replace the 5x^2...
    So it looks like u substitution will not work.


    So then I thought I might be able to use the product rule.
    So I did:

    SA= ∫2[itex]\pi[/itex][(1+ 5x^2)(1/2(100x^2)-1/2)*200x + ((1+100x2)1/2)(10x))]dx |0 to 5


    But when I plug in 5 and 0 from here, I get a large number.
    The number I get is 7521[itex]\pi[/itex].

    But when I put this integral into Wolfram Alpha, it says the answer is an even larger number: 49,894!

    I am confused as to what I am doing wrong!
    Please help!
    Thank you so much! :)
     
  2. jcsd
  3. Feb 10, 2013 #2

    Dick

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    Re: "Find the area of the surface of the curve obtained by rotating th

    There is no real product rule for integrals, I'm not sure what you are thinking there. But to do the integration you need a trig substitution. Like 10x=tan(t).
     
  4. Feb 10, 2013 #3
    Re: "Find the area of the surface of the curve obtained by rotating th

    =_=! Oh wow, I don't know what I'm thinking either! Of course product rule can't be used for integrals! Ugh. wow.

    Okay, so you say that a trig substitution has to be used.
    But how can we replace 10x with tan(x)?

    So do you mean that in (√1 + (10x)2), we replace the 10x with tan(x)?

    ...And how would that help us?
    Thanks!
     
  5. Feb 10, 2013 #4

    Dick

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    Re: "Find the area of the surface of the curve obtained by rotating th

    You don't just replace 10x with tan(x)! This is u-substitution, tan(u)=10x. The sqrt(1+100x^2)=sqrt(1+tan(u)^2)=sqrt(sec(u)^2)=sec(u). And, of course, you also need to figure out how du is related to dx. The usual substitution routine.
     
  6. Feb 11, 2013 #5
    Re: "Find the area of the surface of the curve obtained by rotating th

    Hmmmm.

    I think I see what you mean now about the tan(u)=10x for the u substitution.

    tan(u)=10x

    u= arctan(10x)

    du=(1/((10x)2 + 1)*10dx

    du= 10/(100x2 + 1)dx

    I need to make the dx side have an x^2!

    (100x2 + 1)du = 10dx

    (1 + (1/100x2)du = (10/100x2)dx

    50x4*(1 + (1/100x2))du = (1/10x2)dx *50x4

    (50x4 + 1/2x2)du = 5x2dx

    Well, I got a 5x2 on one side, but this does not help me at all because I still have x's on the other side! X(

    Ugh, please help me here. :/
    I don't see how I can do a u substitution.
    _____________________________________________________

    ∫0 to 5 of 2[itex]\pi[/itex][(1 + 5x2)(√(1 + (10x)2))]dx

    = ∫2[itex]\pi[/itex][(1 + 5x2)(√(1 + (tan(u))2))]dx

    So, I haven't finished my u substitution because I cannot get it to work...
    Thanks for the help!
     
    Last edited: Feb 11, 2013
  7. Feb 11, 2013 #6
    Re: "Find the area of the surface of the curve obtained by rotating th

    And it looks the 49,894 answer I got from Wolfram Alpha is wrong...
     
  8. Feb 11, 2013 #7

    Dick

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    Re: "Find the area of the surface of the curve obtained by rotating th

    No, it's not wrong. You just aren't following through with whole program. You were supposed to convert the integral to a du integral. If tan(u)=10x then sec^2(u)du=10dx. It's maybe somewhat easier if you substitute 10x=sinh(u) if you know hyperbolic trig functions. But it's not supposed to be a very easy problem.
     
    Last edited: Feb 11, 2013
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