       Re: mathematica problem

• To: mathgroup at smc.vnet.net
• Subject: [mg7592] Re: [mg7540] mathematica problem
• From: Allan Hayes <hay at haystack.demon.co.uk>
• Date: Thu, 19 Jun 1997 03:13:55 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```On 13 Jun 1997
Stephanie Gill <sgill at winnie.fit.edu>
in [mg7540] mathematica problem
wrote

> Consider the region enclosed by y=sin^-1x, y=0, and x=1.  Find the
> volume of the solid generated by revolving the region about the
> x-axis using
> a)  disks;
> b)  cylindrical shells.
>
>I do not understand how to do this on Mathematica, can you help?

Stephanie:

Here is the graph of  y = ArcSin[x]   (that is y = sin^-1x) from x
= 0 to x = 1.

In:=
Plot[ArcSin[x],{x,0,1},
Epilog->
{{PointSize[.02],Point[{.7, ArcSin[.7]}],
Text[{"x","y"},{.7, ArcSin[.7]},{-1,1}],
Line[{{1,0},{1, ArcSin}}]
}}
]

You can look at the curve as as given by either
(1)  {x,y} such that y = ArcSin[x]  for x in [0,1]
or
(2)  {x,y} such that x = Sin[y]     for y in [0,ArcSin]

Using (1) we get the volume from disks:

In:=
Integrate[Pi*ArcSin[x]^2, {x, 0, 1}]
Out=
1/4*Pi*(-8 + Pi^2)

Using (2) we get the volume from shells:

In:=
Integrate[2*Pi*y*(1 - Sin[y]), {y, 0, ArcSin}]
Out=
1/4*Pi*(-8 + Pi^2)

This will look clearer if you convert the input and output cells
for the integrations to TraditionalForm (to do this, select the

If you are interested in writing an fuller explanation of what is
happening then Mathematica can produce graphics to show the disks
and shells - please get in touch if you have any further questions

Allan Hayes
hay at haystack.demon.co.uk
http://www.haystack.demon.co.uk/training.html
voice:+44 (0)116 2714198
fax: +44 (0)116 2718642
12 Copse Close, Leicester, LE2 4FB, UK

```

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