homeschool0123
22.08.2019 •
Mathematics
Find the length of the loop for the given curve:
x = 12t - 4t^(3) , y = 12t^(2)
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Ответ:
The arc length,S is determined by
∫ √[(dx/dt)^2 + (dy/dt)^2] dt
= 2 ∫ √[(12 - 12t^2)^2 + (24t)^2] dt from t = 0 to t = 1/√3
= 2 ∫ 12 √[(1 - t^2)^2 + (2t)^2] dt from t = 0 to t = 1/√3
= 24 ∫ √(1 + 2t^2 + t^4) dt from t = 0 to t = 1/√3
= 24 ∫ √(1 + t^2)^2 dt from t = 0 to t = 1/√3
= 24 ∫ (1 + t^2) dt from t = 0 to t = 1/√3
= 24(t + t^3/3) from t = 0 to t = 1/√3
= 24 * (1/√3) * (1 + 1/9)
= 24 * (√3/3) * (10/9)
S = (80/9)√3
Ответ:
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