# Surface area of a curve about the y axis

• Apr 27th 2013, 06:26 AM
OldMate
Surface area of a curve about the y axis
the equation is https://fastres01.qut.edu.au/webwork...0c96109db1.png. 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.
• Apr 27th 2013, 06:45 AM
Prove It
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*}
• Apr 27th 2013, 07:39 PM
OldMate
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?
• Apr 27th 2013, 08:08 PM
Prove It
Re: Surface area of a curve about the y axis
Can you show us your working please?
• Apr 27th 2013, 09:46 PM
OldMate
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
• Apr 27th 2013, 11:18 PM
Prove It
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.
• Apr 28th 2013, 11:47 PM
OldMate
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