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Math Help - Surface area of a curve about the y axis

  1. #1
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    Surface area of a curve about the y axis

    the equation is . I found that x=sqrt(1-y) and that x'=-1/(2*sqrt(y-1)).

    I've been using integral(2*pi*x*sqrt(1+x'^2)) but i keep just getting mixed up with my calculations.

    help would be appreciated.
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    Re: Surface area of a curve about the y axis

    If you rotate this function about the y-axis, each cross-section parallel to the x-axis will be a circle. The circumference of each circle is \displaystyle \begin{align*} 2\pi r = 2\pi x = 2\pi \, \sqrt{ 1 - y } \end{align*}, and if you add up all these circumferences over \displaystyle \begin{align*} 0 \leq y \leq 1 \end{align*} then you will get the total surface area. So

    \displaystyle \begin{align*} SA &= \int_0^1{2\pi \, \sqrt{ 1 - y} \, dy} \end{align*}
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    Re: Surface area of a curve about the y axis

    so i used that equation and my end result was (4pi)/3, but its apparently incorrect. did i do something wrong?
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    Re: Surface area of a curve about the y axis

    Can you show us your working please?
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    Re: Surface area of a curve about the y axis

    I put 2pi out the front and integrated sqrt(1-y) and got -2/3 (1-y)^3/2. when i subbed in 1 I got 0 and when i subbed in 1 i got -2/3, and 0 - -2/3= 2/3. and then multiplied by 2pi
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    Re: Surface area of a curve about the y axis

    I agree with your answer and solution, so I'm not sure what's gone wrong.
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    Re: Surface area of a curve about the y axis

    when you wrote y between 0 and 1, did you actually mean x perhaps cause that is what the question asks. its the only thing i can think of for it not too work
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