Integration by the least$-$square method

The least square method is a graphical method of integration, based on computing the surface located below a function $($which represents the integral of the function$).$ Let us

assume that we have a function $f(x)$ we want to integrate between the two values A and $B.$ If you compute $f(A)$ and $f(B),$ and do the following calculus $f(A)+f(B)*\frac{B-A}{2},$

you can see that you are computing the area of the trapeze defined by $A,f(A),f(B),$ and $B.$ This trapeze will be "close" to the value of the integral of the function if A and $B$ are

very close values $($this is called a linear regression of the function f $).$ So$,$ if you have plenty of small intervals between A and $B,$ you will have a good approximation of the real

integral.

Task $1$

Create a function named surface which will compute the calculus of the surface of a trapeze. The function will take as entry parameter $4$ numbers, and will return one number.

The first line is def surface$(i,j,k,l$ :

Task $2$

We will suppose we have a lower limit for the integral and an upper limit for the integral, and we want to define N intervals in between $($ N can be $1$ or above$).$

Create two functions: one creating the lower limits of each interval, called $I(A,B,N),$ and one creating the upper limit of each interval, called ul$(A,B,N$

$[]$: # Do not change this celli $=0$

j $=$ np$.$pi

k$\}=$n$\backslash$mp@code$\{$np$.$sin$($i$)$

l $=$ np$.$sin$($j$)$

toto $=$ surface$($i$,$j$,$k$,$l$)$

tata $=$ ll$($i$,$j$,100)$

titi $=$ ul$($i$,$j$,100)$