Proving the Volume of a Sphere using the Shell Method. (1 Viewer)

Gussy Booo · Apr 26, 2010

Can someone please do this step by step. Giving explanation as well.

Thanks in advanced.

Drongoski · Apr 26, 2010

$Vol\ of\ sphere\ = \ rotating\ semi-circle:\ \ x^{2}+y^{2}\ =\ r^{2}\ about\ the\ x-axis.\\ \\ V\ \ =\ \ \int_{-r}^{r}\ \pi y^{2}dx\ \ =\ \ \pi \int_{-r}^{r}(r^{2}-x^{2})dx\ \ =\ \ \pi \left [ r^{2}x-\frac{x^{3}}{3} \right ]^{r}_{-r}\\ \\ =\ \ \pi \left [ (r^{2}r-\frac{r^{3}}{3})-(r^{2}(-r)-\frac{(-r)^{3}}{3}) \right ]\ \ =\ \ \frac{4}{3}\pi r^{3}$

Edit:

Thanks duckcowhybrid for pointing out; didn't notice the spec for the shell method; anyway in this case isn't the washer method so much more straightforward.

I find it annoying when an exam question ties you to a certain method when another method gets you home more quickly.

fullonoob · Apr 28, 2010

i think drongoski said it all. rotating semi circle, circle = same shape, could make x the subject and do dy as well

duckcowhybrid · Apr 28, 2010

Drongoski said:
$Vol\ of\ sphere\ = \ rotating\ semi-circle:\ \ x^{2}+y^{2}\ =\ r^{2}\ about\ the\ x-axis.\\ \\ V\ \ =\ \ \int_{-r}^{r}\ \pi y^{2}dx\ \ =\ \ \pi \int_{-r}^{r}(r^{2}-x^{2})dx\ \ =\ \ \pi \left [ r^{2}x-\frac{x^{3}}{3} \right ]^{r}_{-r}\\ \\ =\ \ \pi \left [ (r^{2}r-\frac{r^{3}}{3})-(r^{2}(-r)-\frac{(-r)^{3}}{3}) \right ]\ \ =\ \ \frac{4}{3}\pi r^{3}$

That's by slices though I think. He asked for Shells.

Trebla · Apr 28, 2010

nikkifc · May 1, 2010

Contrary what people have done here, as far as I know, in the HSC you need to sum infinitely small shells or slices and then turn it into an integral...

duckcowhybrid · May 1, 2010

nikkifc said:
Contrary what people have done here, as far as I know, in the HSC you need to sum infinitely small shells or slices and then turn it into an integral...

That's pretty much what they're doing, but they're skipping the lim dx->0 SIGMA from y=0 to y=r as this is basically equivalent to the integral sign, and it's a pain to type in latex. I still write it though.

Proving the Volume of a Sphere using the Shell Method. (1 Viewer)

Gussy Booo

Mathematics <3

Drongoski

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fullonoob

fail engrish? unpossible!

duckcowhybrid

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Trebla

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nikkifc

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duckcowhybrid

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