code for computing area by varying the value of y in matrix?

For computation of area the value of y1,y2 and y3 are coming from three different same dimension matrix and so I want to compute area of using this code for same matrix dimension as y1, y2 and y3?

    x=[7.8 8.25 8.55];
    y=[0.96 0.99 0.94];
    p=polyfit(x,y,2);
    t=linspace(x(1),x(3));
    yhat=polyval(p,t);
    a1=trapz(t,yhat);

How can I improve code for all 3*3 matrix of given y1,y2 and y3, result will be area a1 of 3*3 matrix.

Answer 1

[Roger Stafford]

This general formula in terms of x1, x2, x3, y1, y2, and y3 gives the exact area under your 'polyfit' curve without the trapezoidal approximation errors introduced by 'trapz'. That is, it is the exact integral from x = x1 to x = x3 of the unique quadratic function which passes through all three points (x1,y1), (x2,y2), and (x3,y3).

 a = (x3-x1)*((y1+y2+y3)/3+((x3-x2)*(y2-y1)/(x2-x1)-(x2-x1)*(y3-y2)/(x3-x2))/6);

[Roger Stafford]

A somewhat simpler but equivalent formula for the area is the following:

 a = (x3-x1)*((y1+y3)/2-(x3-x1)*((y3-y2)/(x3-x2)-(y2-y1)/(x2-x1))/6);

[RS]

and I also want to know how can I read y1, y2 and y3 from three different files keeping x1,x2 and x3 as

x=[7.8 8.25 8.55];

[RS]

The value of x1, x2, x3 are coming from different matrices how can I read?

[RS]

How can I read value of several x point from different matrix which are from different matrix?