Find the exact length of the curve.
x = 1 + 9t2
y = 2 + 6t3
0 ≤ t ≤ 2

6*( 5^1.5 - 1 )        

Step-by-step explanation:

Given:-

- The curve is defined by the following two parametric equations:

                                   x = 1 + 9t^2

                                   y = 2 + 6t^3

- The interval for which the required length of curve is defined is [ 0 , 2 ].

Find:-

Find the exact length of the curve

Solution:-

- The length of the curve expressed in the form of time domain parametric equation can be determined from the following integral for length " L ":

- Where the limits of integral are domain for the parameter t = [ 0 , 2 ].

- Compute the first derivative of both parametric coordinates ( x & y ) with respect to parameter ( t ).

- Substitute the respective derivatives in the integral for calculating the length " L " of the curve:

- Apply the general power rule of integration and evaluate the limits:

The exact length of the curve is L = 6*( 5^1.5 - 1 )