Assuming that you know two points (x i , f(x i )) and (x i+1 , f(x i+1 )) , prov

Question

Assuming that you know two points (xi, f(xi)) and (xi+1, f(xi+1)), prove the trapezoidal rule formula

where h = xi+1 - xi. For this, interpolate f(x) as a linear function in the interval [xi, xi+1] and then calculate the

integral of the interpolating polynomial.

Assuming that you know two points (xi, f(xi)) and (xi+1, f(xi+1)), prove the trapezoidal rule formula f(x) dx = h / 2 [f(xi) + f(xi + 1)] where h = xi+1 - xi. For this, interpolate f(x) as a linear function in the interval [xi, xi+1] and then calculate the integral of the interpolating polynomial.

Explanation / Answer

let f(x) = ax+b

then the LHS - (ax^2/2+bx+c) with top and bottom limits xi+1, xi

=>

LHS = a/2(xi+1^2 - xi^2) + b(xi+1-xi) = (xi+1-xi)(axi+1 + axi +2b)/2 = h/2*(axi+1 +b +axi+b) = h/2(f(xi+1)+f(xi))

thus proved

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