Monte Carlo integration of functions to the 3rd power

I have to create a monte carlo integration of the function above and this is what i've tried so far;

function monte_carlo = Task_2_monte_carlo(t)
f = @(t) (-0.1).*(t.^3) + (0.58).*(t.^2).*(cosh((-1)./(t+1))) + exp((t)./(2.7)) + 9.6;
n = rand(10,1);     %deze waardes moeten in de domain van f passen
upper_lim = 12;
lower_lim = 0;
y_max = 200;
N = 1000; %total number of points
N0 = 66;  %number of points underneath the graph
y = (10/n).*(y_max.*(upper_lim - lower_lim))
monte_carlo = sum(y);
plot(n,y); hold on;
end 

This gives me 2 straight lines but I am struggeling to translate this into the correct monte carlo formation. I have succeeded in creating a random graph of poitns which one would need but can't seem to get it coded into the function above, namely;

function monte_carlo = Task_2_monte_carlo_2(t)
f = @(t)(-0.1).*(t.^3) + (0.58).*(t.^2).*(cosh((-1)./(t+1))) + exp((t)./(2.7)) + 9.6;
%yt = sort(rand(2, N)), 1, 'descend'); 
%t = 12*yt(1.:);

n = rand(10,1);     %deze waardes moeten in de domain van f passen
N = 1000; %total number of points
t = 12*rand(1, N); %domain x = [0,12]
y = 200*rand(1, N); %y = [0,200]
plot(t,y,'.')
y_max = 200; 
%N0 = ;  %number of pointa underneed the graph
y1 = (n_max/n).*(y_max.*(max(t) - min(t)))
monte_carlo = sum(y1);
end 

If someone can help or point me in the right direction it would be much appreciated!

Answer 1

[Alan Stevens]

Here's a simple MC calculation of the integration of your function

f = @(t) -0.1*t.^3 + 0.58*t.^2.*cosh(-1./(t+1)) + exp(t/2.7) + 9.6;
t = 0:0.1:12;
% Always a good idea to plot the function first
plot(t,f(t)),grid
%The function values lie below 20, so scatter random points 
% with t values between 0 and 12, and y values between 0 and 20.
% The total rectangular area is
Arect = 12*20;

% The approximate area under the curve is given by

N = 10^4;               % Number of MC trials
t = 12*rand(N,1);       % Random values of t
y = 20*rand(N,1);   % Random values of y
% Now find values of y that are less than or equal to the function.
ylo = y(y<=f(t)); 

% Approximate area
A = Arect*numel(ylo)/N;
disp(A)

(This could be made more efficient by noticing that the curve doesn't drop below 5 over the domain of interest.)

[Denise Veldhuis]

Thank you so much! I changed some parts of it but I see what I was messing up, thank you!