lijnstukje = \sqrt{\Delta x^2+\Delta y^2}

= \sqrt{\Delta x^2 + \Delta y^2 \cdot\frac{\Delta x^2}{\Delta x^2}

= \sqrt{\Delta x^2 + \frac{\Delta y^2}{\Delta x^2} \cdot\Delta x^2}

= \sqrt{\Delta x^2 + \left(  \frac{\Delta y}{\Delta x} \right)^2\cdot\Delta x^2}

=\sqrt{ \Delta x^2 \cdot \left( 1 + \left( \frac{\Delta y}{\Delta x} \right)^2 \right)}

=\Delta x \cdot \sqrt{  1 + \left( \frac{\Delta y}{\Delta x} \right)^2 }

booglengte = \sum_{k = 1}^{n}{ \Delta x \cdot \sqrt{  1 + \left( \frac{\Delta y_k}{\Delta x} \right)^2 } }

= \lim_{ \Delta x \to 0 }\sum_{k = 1}^{n}{ \Delta x \cdot \sqrt{  1 + \left( \frac{\Delta y_k}{\Delta x} \right)^2 } }

= \int_{x_A}^{x_B}{ \sqrt{  1 + f'(x)^2 }  } dx

n = 1 (slider from 1 to 100, step 1)
f(x) = (0.1 * (x - 10)^(3)) + (x - 10)^(2) + x - 10 + 3
x_B = Point(xAxis)
B_x: x(x_B)
B: (x(x_B), f(x(x_B)))
x_A = Point(xAxis)
A_x: x(x_A)
A: (x(x_A), f(x(x_A)))
a = Segment(x_A, A)
b = Segment(x_B, B)
f_b = Function(f, A_x, B_x)
Booglengte = Length(f, A_x, B_x)
tekst1: "Door GeoGebra berekende booglengte = " + Booglengte + ""
DX: (B_x - A_x) / n
AB: sqrt((x(B) - x(A))^(2) + (y(B) - y(A))^(2))
Ys = Sequence(f(A_x + (p * DX)), p, 0, n, 1)
Xs = Sequence(x(x_A) + (p * DX), p, 0, n, 1)
Ps = Sequence((Element[Xs, p], Element[Ys, p]), p, 1, n + 1, 1)
Hs = Sequence(Segment[Element[Ps, p], Element[Ps, p + 1]], p, 1, n, 1)
DXs = Sequence(Segment[Element[Ps, p], (x(Element[Ps, p + 1]), y(Element[Ps, p]))], p, 1, n, 1)
DYs = Sequence(Segment[Element[Ps, p + 1], (x(Element[Ps, p + 1]), y(Element[Ps, p]))], p, 1, n, 1)
Bs = Sequence(sqrt(DX^(2) + (Element[Ys, p + 1] - Element[Ys, p])^(2)), p, 1, n, 1)
BooglengteBenadering = Sum(Bs)
tekst2: "Benadering met lijnstuk = " + AB + ""
tekst3: "Benadering met " + n + " lijnstukjes = " + BooglengteBenadering + ""