Final answer:
The length of the curve given by the parametric equations x=12t^3 and y=18t^2 from t=0 to t=\sqrt{8} can be determined using the arc length formula for parametric curves, by integrating the square root of the sum of the squares of the derivatives of x and y with respect to t.
Step-by-step explanation:
To find the length of the curve with the given parametric equations x=12t^3 and y=18t^2 from t=0 to t=\sqrt{8}, we need to use the formula for the arc length of a curve defined by parametric equations:
The formula for the arc length, L, is:
L = \int from a to b \sqrt((dx/dt)^2 + (dy/dt)^2) dt
First, we calculate the derivatives of x and y with respect to t:
dx/dt = 36t^2
dy/dt = 36t
Now we plug these into the arc length formula and integrate from 0 to \sqrt{8}:
L = \int from 0 to \sqrt{8} \sqrt((36t^2)^2 + (36t)^2) dt
This integral will yield the length of the curve.