Simpson's Rule

Date: 04/12/97 at 16:11:25
From: Adam
Subject: Using Simpson's Rule to find volume

Hi,

Recently at school we came across a question which asks you to find 
the volume when a curve is rotated about the x-axis, using Simpson's 
rule.  Could you please solve it and include a worked solution and 
theory behind your answer?

Q. Find the volume of the solid formed if the curve  y = cos -1(x)   
is rotated about the x-axis between x=0 and x=1 (use Simpson's Rule 
with 3 function values).

Note: y = cos -1(x) means y = cos inverse of x.
(cos to the power of -1)

Thanks in advance.

Date: 04/13/97 at 11:51:05
From: Doctor Anthony
Subject: Re: Using Simpson's Rule to find volume

The volume of revolution is given by INT[pi.y^2.dx].

So to use Simpson's Rule, we use the usual formula, but with ordinates 
y^2 instead of y, and also multiply by pi.

The curve is the arccos curve - also denoted by cos^(-1)(x).

With only three function values we have x=0, x=1/2 and x=1
The table of values is as shown below:

    x       y = cos^(-1)(x)    y^2
   ---------------------------------
    0           pi/2          2.4674 = y0
   0.5         1.047          1.0966 = y1
    1            0              0    = y2

Simpson's Rule then gives: (using h = 1/2 and the factor pi in front).

   V = pi/6[y0 + 4y1 + y2]
     = pi/6[2.4674 + 4 x 1.0966 + 0]
     = pi/6[6.8538]
     = 3.5886.
    
-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/