%%%    Convergence Team Lab  Solution

%% Problem 1.

%% Area of Sphere
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function area = areaSphere( x) 

% This function computes the area of a half a cross section of a sphere.
rad = sqrt( 9 - x.^2) ;
area = 0.5*pi*rad.^2 ;

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%% Volume of a Sphere 

function vol = volumeSphere( X )

% This function computes the volume of the sphere from -3 to X.

vol = quad( 'areaSphere', -3, X ) ;

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% When is volume = 20?  In preparation for later sections.
function z = tobezeroSphere( X) 

%  This is the function passed to fzero.

z = volumeSphere(X) - 20 ;

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%  To find the value of x where the volume is 20, use
x = fzero( 'tobezeroSphere', -2, 2 )


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%%  Problem 2

%% Area of Torus

function area = areaTorus2(x)
% This function computes the area of a shell of the torus.

y = sqrt( 1 - (x-2).^2 ) ;
height = 2.*y ;
area = 2*pi*height.*x ;
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function vol = volumeTorus2( x) 

% This function computes the volume of the torus from 1 to X.

vol = quad( 'areaTorus2', 1, x) ;
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% When is volume = 20?  
function z = tobezeroTorus2( x )

% This is the function passed to fzero for Problem 2.

z = volumeTorus2(x) - 20 ;
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%  To find the value of x where the volume is 20, use
x = fzero( 'tobezeroTorus2', 1.8, 2.3 )

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%  Problem 3
function area = areaTorus3(y )
% This function computes the area of the disk for part 3.

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

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

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function vol = volumeTorus3( s )
% This function computes the volume of the torus from -s to s.

vol = 2*quad( 'areaTorus3', 0, s ) ;

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function z = tobezeroTorus3(s)
% This is the function passed to fzero for Problem 3.

z = volumeTorus3(s ) - 20 ;

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%  To find the value of x where the volume is 20, use
x = fzero( 'tobezeroTorus3', 0, .8 )