Hey,
I'm currently doing the volumes of solids topic and the question is:
By taking slices work out the volume of the area enclosed within the circle (x-1)^2 + y^2 = 1 rotated about the y-axis
I'm not getting to the answer and I don't know where the mistake is. I would appreciate some help
Thank you 
Fe
Hey there aoifera! Welcome to the forums!
That's a bit of a nasty one you've got there, but let's have a look
using geometric formula for the volume of a horn torus, I'm expecting an answer of 2 pi squared, so let's see how I go
The volume of the solid when you rotate the circle about the
y-axis is interesting. What we should do is separate the circle into two halves:
This separates the circle into the two halves, left and right. The volume when we rotate the circle around the y axis is just the
volume when we rotate the right hand side, minus the volume when we rotate the left hand side. So, we set up our formula like so;
^2-(1-\sqrt{1-y^2})^2dy \\ = 4\pi\int^1_{-1}\sqrt{1-y^2}dy )
At this point we should use symmetry to simplify slightly (which we can do because this is an even function we are integrating):
Now we can use a substitution here, the easiest is probably:
Or, you can consider the integral geometrically. The area under the curve from 1 to 0 is just a quarter of the area of a unit circle. Whichever method you choose:
I hope this helps!! It might be ever so slightly different to how the slices method usually works, I didn't do Extension 2 in the HSC, I'm just doing the question how I'd do it at uni
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