# Answer to Question #2208 in Calculus for lune

Question #2208

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

Expert's answer

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=cos

y=sin3t 0 1 0 -1 0 1 0

d=√((x

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

t 0 π/6 π/3 π/2 2π/3 5π/6 π

x=cos

^{2}t 1 3/4 1/4 0 1/4 3/4 1y=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 √17/4The 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

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