Each of these Matlab functions must be defined in its own .m file.
They are included here in one text file for easier viewing by students.


==============================================================================
  areaHemisphere.m
==============================================================================
function area = areaHemisphere( x ) 

% The areaHemisphere function computes the area of a cross-section of 
% a hemisphere. 

radius = sqrt( 9 - x.^2 ) ;
area = 0.5*pi*radius.^2 ;



==============================================================================
  areaTorus2.m
==============================================================================
function area = areaTorus2( t )

% The areaTorus2 function computes the cross-sectional area of the torus
% at x = t using the method of shells.

y = sqrt( 1 - (t-2).^2 ) ;
height = 2.*y ;
area = 2*pi*height.*t ;



==============================================================================
  volumeTorus2.m
==============================================================================
function vol = volumeTorus2( t ) 

% The volumeTorus2 function computes the volume of the torus from 
% x = 1 to x = t.

vol = quad( @areaTorus2, 1, t ) ;



==============================================================================
  tobezeroTorus2.m
==============================================================================
function z = tobezeroTorus2( t )

% The tobezeroTorus2 function computes the volume of the torus from
% x = 1 to x = t and then subtracts 20 from that value.  This function
% can be passed to Matlab's fzero in order to determine where the volume
% of the cut-off torus is equal to 20 (for Problem 2.d).

z = volumeTorus2(t) - 20 ;



==============================================================================
  areaTorus3.m
==============================================================================
function area = areaTorus3( t )

% The areaTorus3 function computes the cross-sectional area of the torus
% at y = t using the method of washers.

rad1 = 2 - sqrt( 1 - t.^2 ) ;
rad2 = 2 + sqrt( 1 - t.^2 ) ;

area = pi*( rad2.^2 - rad1.^2 ) ;



==============================================================================
  volumeTorus3.m
==============================================================================
function vol = volumeTorus3( t )

% The volumeTorus3 function computes the volume of the torus from 
% y = -t to y = t.

vol = 2*quad( @areaTorus3, 0, t ) ;



==============================================================================
  tobezeroTorus3.m
==============================================================================
function z = tobezeroTorus3( t )

% The tobezeroTorus3 function computes the volume of the torus from
% y = -t to y = t and then subtracts 20 from that value.  This function
% can be passed to Matlab's fzero in order to determine where the volume
% of the cut-off torus is equal to 20 (for Problem 3).

z = volumeTorus3(t) - 20 ;