Final answer:
The exact length of the curve y = 5 + 6x^(3/2), 0 ≤ x ≤ 1, is approximately 81.087 units.
Step-by-step explanation:
To find the exact length of the curve, we can use the formula for arc length: L = ∫sqrt(1 + (dy/dx)^2) dx, where dy/dx represents the derivative of y with respect to x. In this case, y = 5 + 6x^(3/2), so dy/dx = 9x^(1/2). We can substitute these values into the formula and integrate over the interval 0 ≤ x ≤ 1:
L = ∫sqrt(1 + (9x^(1/2))^2) dx
L = ∫sqrt(1 + 81x) dx
Now, we can evaluate the integral:
L = ∫(1 + 81x)^(1/2) dx
Using the power rule for integration, we get:
L = (2/3)(1 + 81x)^(3/2) + C
Plugging in the bounds of the interval, we find:
L = (2/3)(1 + 81)^(3/2) - (2/3)(1 + 81(0))^(3/2)
L = (2/3)(82)^(3/2) - (2/3)(1)^(3/2)
L = (2/3)(82)^(3/2) - 2/3
L ≈ 81.087