Final answer:
To find the exact length of the curve, we can use the formula for arc length and integrate the expression.
Step-by-step explanation:
To find the exact length of the curve, we can use the formula for arc length, which is given by:
L = ∫[a, b] √[1+(dy/dx)²] dx
In this case, we have the parametric equations x = 7 + 6t² and y = 6 + 4t, with t ranging from 0 to 2. To find dy/dx, we differentiate y with respect to x:
dy/dx = (dy/dt) / (dx/dt)
Substituting the given equations, we get:
dy/dx = (4) / (12t)
Now we can substitute dy/dx into the formula for arc length and integrate:
L = ∫[0, 2] √[1 + (4/12t)²] dt
Integrating this expression is challenging, but it can be done using trigonometric substitution. The final result for the exact length of the curve is not easily expressed in a simple form, but it can be approximated using numerical methods.