Find the length of the curve. Select the correct answer.
$$x = 3 t ^ { 2 } + 8 , y = 2 t ^ { 3 } + 8,0 \leq t \leq 1$$
A) $2 \sqrt { 2 } - 2$
B) $4 \sqrt { 2 }$
C) $4 \sqrt { 2 } - 1$
D) $4 \sqrt { 2 } - 2$
E) None of these

Question

Quizplus · Accepted Answer

The length of the curve is calculated using the formula $\int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$. Here, $\frac{dx}{dt} = 6t$ and $\frac{dy}{dt} = 6t^2$. The integral becomes $\int_{0}^{1} \sqrt{(6t)^2 + (6t^2)^2} \, dt = \int_{0}^{1} 6t\sqrt{1 + t^2} \, dt$. Solving this integral gives the length $4\sqrt{2} - 2$.

Find the Length of the Curve x=3t2+8,y=2t3+8,0≤t≤1x = 3 t ^ { 2 } + 8 , y = 2 t ^ { 3 } + 8,0 \leq t \leq 1x=3t2+8,y=2t3+8,0≤t≤1

Find the Length of the Curve $x = 3 t ^ { 2 } + 8 , y = 2 t ^ { 3 } + 8,0 \leq t \leq 1$