easy volume (1 Viewer)

shuning · May 14, 2009

the curve $y^2=x$ and the line x =4 revolved about the line y=2

thx

Drongoski · May 14, 2009

$\100dpi V\ =\ \pi \int_{0}^{4}\left [ (-\sqrt{x}-2)^{2}\ -\ (\sqrt{x}-2)^{2} \right ]\ dx\\ \\ \ =\ \ \pi \int_{0}^{4}\left [ (x+4\sqrt{x}+4)\ -\ (x-4\sqrt{x}+4) \right ]\ dx\\ \\ \ =\ \ 8\pi \int_{0}^{4}\ \sqrt{x}\ dx\\ \\ \ =\ \ 8\pi \times \frac{2}{3}x^{\frac{3}{2}}]^{4}_{0}\ \ =\ \ \frac{16\pi }{3}(8\ -\ 0)\ \ =\ \ \frac{128\pi }{3}\ cubic\ units$

Edit

My answer was wrong: should be 128 pi/3. above now amended.

shuning · May 14, 2009

回复: Re: easy volume

dont fully get it.... u used R - r right?

GUSSSSSSSSSSSSS · May 14, 2009

Re: 回复: Re: easy volume

shuning said:
dont fully get it.... u used R - r right?

yea he did

Drongoski · May 14, 2009

Re: 回复: Re: easy volume

shuning said:
dont fully get it.... u used R - r right?

There are 2 branches to curve: y = sqr(x) and y = -sqr(x)

The lower one on rotating about y=2 has radius -sqr(x) - 2; upper one radius sqr(x)-2 forming a washer of thickness delta x which you integrate over [0,4]

The bigger radius generates a disc of area pi x (-sqr(x) - 2)² and the smaller radius a circle of area pi x (sqr(x) - 2)² and their difference is the area of the washer.

GUSSSSSSSSSSSSS · May 14, 2009

Re: 回复: Re: easy volume

drongoski u wudnt consider doing this by shells ??

....normally whenever i see a washer/annulus type shape i normally automatically think about doin it by shells first ....unless u gotta split it up xD

Drongoski · May 14, 2009

Re: 回复: Re: easy volume

GUSSSSSSSSSSSSS said:
drongoski u wudnt consider doing this by shells ??

....normally whenever i see a washer/annulus type shape i normally automatically think about doin it by shells first ....unless u gotta split it up xD

You can do it by shells of course. I happen to have done a similar one recently this way. I therefore felt safer doing it by annulus.

Drongoski · May 15, 2009

Re: 回复: Re: easy volume

GUS . . . SSS

Volume by shell method

$\100dpi V\ \ =\ \ \int_{-2}^{2}\ 2\pi (2-y)\times (4-x)\ dy\ \ =\ \ 2\pi \int_{-2}^{2}(8-2x-4y+xy)\ dy\\\\ \ \ =\ \ 2\pi \int_{-2}^{2}(8-2y^{2}-4y+y^{3})\ dy\ \ =\ \ 2\pi \left [ 8y-\frac{2}{3}y^{3}-2y^{2}+\frac{1}{4}y^{4} \right ]^{2}_{-2}\\ \\ \ =\ \ 2\pi \left [ 8y-\frac{2}{3}y^{3} \right ]^{2}_{-2}\ \ =\ \ 2\times 2\pi (8\times 2- \frac{2}{3}\times 2^{3})\ \ =\ \ 4\pi \times \frac{32}{3}\ \ =\ \ \frac{128\pi }{3}\ \ unit^{3}$

For this question, which of the 2 methods would you prefer ??

GUSSSSSSSSSSSSS · May 15, 2009

Re: 回复: Re: easy volume

lol well the algebra worked out VERY nicely for doin it by slices with EVERYTHING cancelling out lol

....i guess it's fairly similar for both methods with a shape like that, its when u get them donut thingies...torus? ....that shells pwns slices xD

badquinton304 · May 16, 2009

Re: 回复: Re: easy volume

GUSSSSSSSSSSSSS said:
lol well the algebra worked out VERY nicely for doin it by slices with EVERYTHING cancelling out lol

....i guess it's fairly similar for both methods with a shape like that, its when u get them donut thingies...torus? ....that shells pwns slices xD

Yeah, I hate doing doughnut by slice, shells is much easier cause you do not have to use washers or anything.

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easy volume (1 Viewer)

shuning

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Drongoski

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shuning

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GUSSSSSSSSSSSSS

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Drongoski

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