Final answer:
To find the exact length of the curve x = (1/3)(y² + 2)³/² on 2 ≤ y ≤ 6, we can use the arc length formula and integrate.
Step-by-step explanation:
To find the exact length of the curve x = (1/3)(y² + 2)³/² on 2 ≤ y ≤ 6, we can use the arc length formula. The formula for finding the arc length of a curve y = f(x) on the interval (a, b) is given by:
Length = ∫(a to b) sqrt[1 + (f'(x))²] dx
In this case, we have x = (1/3)(y² + 2)³/². To find f'(x), we can take the derivative of x with respect to y and then solve for dy/dx:
dy/dx = (d/dy [(1/3)(y² + 2)³/²]) / (d/dx [(1/3)(y² + 2)³/²])
Once we have dy/dx, we can substitute it into the arc length formula and integrate to find the exact length of the curve.