anonymous
  • anonymous
Find the volume of the torus (doughnut) formed by rotating the circle x2 + (y − 2)2 = 1 about the x-axis.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
Hint: \[\int\limits_{-1}^{1}\sqrt{1-x ^{2}}=\frac{ \pi }{ 2 }\]
klimenkov
  • klimenkov
|dw:1357858021843:dw|
anonymous
  • anonymous
???

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klimenkov
  • klimenkov
Do you know the formula for the volume of solid rotating \(y=f(x)\) about \(x\)-axis?
anonymous
  • anonymous
yes.
klimenkov
  • klimenkov
|dw:1357858353355:dw|
anonymous
  • anonymous
\[\pi \int\limits_{a}^{b}f(x)dx\]
klimenkov
  • klimenkov
You forgot 2.
anonymous
  • anonymous
ma bad
klimenkov
  • klimenkov
|dw:1357858535828:dw| Subtract the right volume from the left. Got it?
anonymous
  • anonymous
y subtracting?
klimenkov
  • klimenkov
|dw:1357858823945:dw| It is like subtracting this areas.
klimenkov
  • klimenkov
\(V=\pi\int_{-1}^1(\sqrt{1-x^2}+2)^2dx-\pi\int_{-1}^1(-\sqrt{1-x^2}+2)^2dx\)
klimenkov
  • klimenkov
If you don't get it now just sit and think about it very thoroughly.
anonymous
  • anonymous
|dw:1357858979556:dw| the graph of x^2+(y-2)^2=1 is something like that how do we get to the penultimate drawing
anonymous
  • anonymous
i kinda get it you reflect a portion of the circle in the x axis and find the region bounded by the two arcs|dw:1357859345394:dw|
klimenkov
  • klimenkov
if you try to write y depending on x, you will get 2 functions: \(y=\sqrt{1-x^2}+2\) and \(y=-\sqrt{1-x^2}+2\). Now think what will be if you rotate them about x-axes. And then use your imagination to unterstand how to get the needed volume. Really the plot is:|dw:1357859397346:dw|
klimenkov
  • klimenkov
But if you are lazy and dishonest student you can just use the formulas I've written above.
anonymous
  • anonymous
thx
klimenkov
  • klimenkov
Have you understood something?
anonymous
  • anonymous
yes it is basically a volume but it complicated by the fact that the function that we get after making y the subject of the formula is composed of two equations so we add the two. right?
klimenkov
  • klimenkov
It is very hard to tell if you got something or not. I don't know your level of knowledge. Tell me honestly: can you find the area between \(y=x^2\) and \(y=1-x^2\)?
anonymous
  • anonymous
subtract x^2 from 1-x^2 and integrate
klimenkov
  • klimenkov
Why square?
anonymous
  • anonymous
juss realised my prob no squaring
anonymous
  • anonymous
integrate, limits the intersection of the lines
klimenkov
  • klimenkov
Yes. The similar situation is here. But now we have a volume to subtract. Hope, you got something, if not - very bad.
anonymous
  • anonymous
so basically in this prob they ask us to find the volume rather than the area
anonymous
  • anonymous
can you pls draw the resultant volume if it is not too diff
klimenkov
  • klimenkov
Yes. The volume of the doughnut.|dw:1357861006645:dw|
anonymous
  • anonymous
thx for your help

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