Volumes - Slicing (1 Viewer)

ashokkumar · May 14, 2010

Use the method of slicing to find the volume of the solid obtained by rotating the region enclosed within the circle (x-1)^2 + y^2 = 1 about the y-axis.

Drongoski · May 14, 2010

ashokkumar said:
Use the method of slicing to find the volume of the solid obtained by rotating the region enclosed within the circle (x-1)^2 + y^2 = 1 about the y-axis.

I thought this was an easy question - so didn't bother. Is answer 2 (pi)^2 cubic units ??

ashokkumar · May 14, 2010

yea it is. i was actually more interested in seeing how other people did it.

Drongoski · May 14, 2010

$Preliminaries:\ Let\ I= 4\pi \int_{0}^{2}\sqrt{1-(x-1)^{2}}\ dx\\ \\ Let\ u=x-1\ \Rightarrow\ dx=du\ \ and\ \ x=0,2\ \Rightarrow \ u=-1,1\\ \\ \therefore \ \ I\ =\ 4\pi \int_{-1}^{1}\sqrt{1-u^{2}}du\ \ =\ \ 4\pi \times area\ of\ \ semi-circle\ radius\ 1\\ \\ =\ \ 4\pi \ \times \ \frac{\pi }{2} \ \ =\ 2\pi ^{2}\ unit^{3}\\ \\ Vol\ of\ revolution\ \ V\ =\ \int_{0}^{2}\ 2\pi x\cdot 2y\ dx\ \ =\ \ \int_{0}^{2}2\pi x\cdot 2\sqrt{1-(x-1)^{2}}dx\\ \\ =\ \ -2\pi \int_{0}^{2}-2x\sqrt{1-(x-1)^{2}}dx\ =\ -2\pi \int_{0}^{2}[-2(x-1)-2]\sqrt{1-(x-1)^{2}}dx\\ \\ =\ -2\pi \int_{0}^{2}\sqrt{1-(x-1)^{2}}\ d(1-(x-1)^{2})\ +\ 4\pi \int_{0}^{2}\sqrt{1-(x-1)^{2}}dx\\ \\ = \ -2\pi \cdot \frac{2}{3}\left [ (1-(x-1)^{2})^{\frac{3}{2}} \right ]_{0}^{2}\ +\ I \ \\ \\ = \ 0\ +\ 2\pi ^{2}\ \ =\ \ 2\pi ^{2}\ unit^{3}$

The 0 for the 1st integral comes from usual working out which I'm too lazy to spell out.

fullonoob · May 16, 2010

3rd line, dno how do xD for vol of revolution
teach teach, how you split it up like that

Drongoski · May 16, 2010

I worked on the problem earlier and found I had to evaluate the integral in my "Preliminaries" as part of whole solution. So I decided to work this one out first so I could use the results later, as I've done in my solution.

or which 3rd line ??

Drongoski · May 16, 2010

ashokkumar said:
yea it is. i was actually more interested in seeing how other people did it.

But how did you do it ???

fullonoob · May 16, 2010

the d(apple) , d(asdasdad) line where you split it , the line above the letter I, wna learn it for the heck of it xD
i do it by convention

Volumes - Slicing (1 Viewer)

ashokkumar

Mr

Drongoski

Well-Known Member

ashokkumar

Mr

Drongoski

Well-Known Member

fullonoob

fail engrish? unpossible!

Drongoski

Well-Known Member

Drongoski

Well-Known Member

fullonoob

fail engrish? unpossible!

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