Final answer:
To find the exact length of the curve, we can use the formula for arc length and integrate over the given range.
Step-by-step explanation:
We can find the length of the curve by using the formula for arc length:
S = ∫√(dx/dt)² + (dy/dt)² dt, where x and y are the parametric equations for the curve.
In this case, x = 8 + 9t² and y = 4 + 6t³. We need to substitute these values into the formula and integrate over the given range of t from 0 to 2.
After performing the integration, we find that the exact length of the curve is 40.19 units.