Q.309. Arc Length of a curve.

Question 309.
Find the exact length of the curve:
y^2 = 4(x+4)^3, 0 ≤ x ≤ 2, y > 0.

Answer 309.
y^2 = 4 (x+4)^3
=> y = 2 (x+4)^(3/2)
=> dy/dx = 3(x+4)^(1/2)
=> (dy/dx)^2 = 9(x + 4)

arc length
= ∫ √[1 + (dy/dx)^2] dx ... [x=0 to x=2]
= ∫ √[1 + 9x + 36] dx ... [x=0 to x=2]
= ∫ √[9x + 37] dx ... [x=0 to x=2]
= (1/9) * (2/3) (9x + 37)^(3/2) ... [x=0 to x=2]
= (2/27) [(55)^(3/2) - (37)^(3/2)]
= (2/27) [407.89 - 225.06]
≈ 13.543.

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