% Convergence Team Lab

% Part 1

%%  Find the volume of two portions of a hemisphere.
%%  The function crossSection(x) computes the cross sectional area.

disp(['  ' ]) %  Get a blank line

X = 0;
volPartSphere = quad('crossSection', -3, X);
disp([' For x = ', num2str(X),' the volume of the piece of the hemisphere is ',...
         num2str(volPartSphere) ])

check = (4/3)*pi*3^3 / 4;
disp([' If everything is correct, then ', num2str(volPartSphere), ...
        ' must equal ', num2str(check)])

X = 1.5;
volPartSphere = quad('crossSection', -3, X);
disp([' For x = ', num2str(X),' the volume of the piece of the hemisphere is ',...
         num2str(volPartSphere) ])


% Part 2
%%  Compute the volume of the torus.
%  Find the volume at t = 2

t = 2;
vol = quad('areaShell', 1, t);
disp([' For x = ', num2str(t),' the volume of the torus is ', num2str(vol) ])

%  Find the volume at t = 3

t = 3;
vol = quad('areaShell', 1, t);
disp([' For x = ', num2str(t),' the volume of the torus is ', num2str(vol) ])

%% Put the volume in a function and do it again.

t = 2;
vol = volumeTorus(t) ;
disp([' For x = ', num2str(t),' the volume of the torus is ', num2str(vol) ])

t = 3;
vol = volumeTorus(t);
disp([' For x = ', num2str(t),' the volume of the torus is ', num2str(vol) ])

%%  Find the value of t so that the volume of that piece of
%%  the torus is 20.

tShell = fzero('volLess20', 2);
disp([' For problem 2.b the volume is 20 when x = ', num2str(tShell) ])


% Part 3

%%  Find the value of t so that the volume of that piece of
%%  the torus is 20.

tSlice = fzero('volDiskLess20', 0.5);
disp([' For problem 3 the volume is 20 when y = ', num2str(tShell) ])function area = crossSection(x)

% This function gives the area of the half circle that
% intersects the hemisphere at x.

radius = sqrt( 9 - x.^2 );

area = 0.5*pi*radius.^2 ;%
% Computes the area of the shell taken by cutting the torus given by
%    (x - 2)^2 + y^2 = 1
% parallel to the y-axis at x.
%
function a = areaShell(x)

% Area of the shell at x is 2*pi*x*ht(x)  
% where ht(x) is the height of the shell at x

height = 2*sqrt(1 - (x - 2).^2);
a = 2*pi*x.*height;%
% Computes the volume of the solid obtained by cutting the torus along
% the x-axis at t.
%
function vol = volumeTorus(t)

vol = quad('areaShell', 1, t);%
% Computes (the volume of the solid obtained by cutting the torus along
% the x-axis at t ) - 20
%
% It's necessary to do this to use the function with Matlab's fzero 
% function (which assumes equations are set up in standard form: 
% f(x) = 0).
%
function getToZero = volLess20(t)

getToZero = volumeTorus(t) - 20;%
% Computes the area of a slice of the torus parallel to the x-axis 
% taken at height y.
%
function a = areaSlice(y)

radiusOuter = 2 + sqrt(1 - y.^2); % radius of outer disk
radiusInner = 2 - sqrt(1 - y.^2); % radius of inner disk

% area = area of outer disk - area of inner disk
a = pi*(radiusOuter.^2 - radiusInner.^2);%
% Computes the volume of the solid obtained by slicing the torus
%      (x - 2)^2 + y^2 =1
% parallel to the x-axis at t and -t.
%
function vol = volumeDisk(t)

% take the volume from 0 to t and double it to get the volume from 
% -t to t
vol = 2 * quad('areaSlice', 0, t);%
% Computes (the volume of the solid obtained by slicing the torus 
% parallel to the x-axis at t and -t) - 20.
%
% It's necessary to do this to use the function with Matlab's fzero 
% function (which assumes equations are set up in standard form: 
% f(x) = 0).
%
function getToZero = volDiskLess20(t)
getToZero = volumeDisk(t) - 20;