Arc length?

x'
= 3*[ - 0.4*e^(-0.4t)*sin 2t + 2*e^(-0.4t)*cos 2t ]
= 3*e^(-0.4t)*( 2*cos 2t - 0.4*sin 2t )

y'
= 3*[ - 0.4*e^(-0.4t)*cos 2t - 2*e^(-0.4t)*sin 2t ]
= - 3*e^(-0.4t)*( 2*sin 2t + 0.4*cos 2t )

∣ v ∣
= ( x' )^2 + ( y' )^2
= 9*e^(-0.8t)*(4+0.16)
= 37.44*e^(-0.8t) ..... Ans of 1st blank

arc length
= ∫ ∣ v ∣ dt , from t = 0 to t = 6
= ∫ 37.44*e^(-0.8t) dt
= 37.44*(-1/0.8)*[ e^(-0.8t) ] , from t = 0 to t = 6
= - (37.44/0.8)*[ e^(-4.8) - 1 ]
≒ 46.4 ..... Ans of 2nd blank