Find the exact length of the curve. x = 8 + 12t2 y = 3 + 8t3 0 ? t ? 5 Solution

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example Find the exact length of the curve. (Calculus Question)? x = 5 + 12t2 y = 5 + 8t3 0 = t = 3 answer 1) As x = 5 + 12(t^2); ==> dx/dt = 24t or dx = 24t dt 2) As y = 5 + 8(t^3), dy/dt = 24(t^2) 3) Hence, dy/dx = (dy/dt)/(dx/dt) = t 4) ==> 1 + (dy/dx)^2 = 1 + t^2 5) Length of a curve is = Int[rt{1 + (dy/dx)^2}]dx in limits (a,b) 6) So here, length = Int[rt(1 + t^2)](24t)(dt) in limits 0 = t = 3 [Substituting for dx = 24t dt from step 1] 7) Let 1 + t^2 = u; ==> 2tdt = du; when t = 0, u = 1 and when t = 3, u = 10 8) ==> Length = 12 x Int[rt(u) du] in limits [1, 10] 9) ==> L = 12 x {(u^3/2)/(3/2)} in [1, 10] ==> L = 8 x [10^3/2 - 1^3/2] ==> L = 8 x [10(rt 10) - 1] This approximately = 245 units.

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