Answer
$\approx 86.631$
Work Step by Step
General formula for the arc length:
$l=\int_m^n \sqrt {(x'(t))^2+(y'(t))^2+(z'(t))^2} dt$
$\implies \int_0^3 \sqrt {(2t)^2+(3t^2)^2+(4t^3)^2} dt$
$\implies \int_0^3 \sqrt {4t^2+9t^4+16t^6} dt$
We will have to write the Simpson Rule for $n+1=7$ as:
$l=\int_0^7 \sqrt {4t^2+9t^4+16t^6} dt $
$\approx \dfrac{\triangle x}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5)+f(x_6)$
This gives: $l\approx \dfrac{0.5}{3}[0+4(1.34629)+2(5.38516)+4(15.38871)+2(34.4093)+4(65.4432)+111.48543]$
Hence, we have $l \approx 86.631$
