Answer to Question #2208 in Calculus for lune

Question

Estimate the arc length of r(t) = (cos^2 t, sin 3t), t element of [0, pi] using trapezoidal rule with n = 6.

Answer 1

Divide the interval in n parts. Find distance between every pair of two nearest points.

t 0 π/6 π/3 π/2 2π/3 5π/6 π
x=cos²t 1 3/4 1/4 0 1/4 3/4 1
y=sin3t 0 1 0 -1 0 1 0
d=√((x_k-x_k-1)²+(y_k - y_k-1 )²) - √17/4 √5/2 √17/4 √17/4 √5/2 √17/4

The arc length is the sum of smaller parts:
L = √17/4 + √5/2 + √17/4 + √17/4 + √5/2 + √17/4 = √17 + √5 ≈ 6.3592