Answer:
4.667
Step-by-step explanation:
f(x) = 2/3 × x^(3/2) + 4
the arc length of a function f(x) between 2 points A and B is
A integral B (sqrt(1 + f'(x)²)dx)
in our case
f'(x) = 3/2 × 2/3 × x^(1/2) = x^(1/2)
f'(x)² = (x^(1/2))² = x^(2/2) = x¹ = x
A = 0
B = 3
0 integral 3 (sqrt(1 + x))dx) = 0 integral 3 ((1 + x)^(1/2) dx)
0 interval 3 (2/3 × (1 + x)^(3/2)) =
= 2/3 × (1 + 3)^(3/2) - 2/3 × (1 + 0)^(3/2) =
= 2/3 × sqrt(4³) - 2/3 × 1 = 2/3 × 8 - 2/3 = 16/3 - 2/3 =
= 14/3 = 4.66666666... ≈ 4.667
