The length of the curve $x ( t ) = t ^ { 3 } , y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ is

A) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$
B) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$
C) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$
D) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$
E) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$

Question

The length of the curve $x ( t ) = t ^ { 3 } , y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ is

A) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$
B) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$
C) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$
D) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$
E) $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$

Quizplus · Accepted Answer

The length of the curve is calculated using the formula $\int_{a}^{b} \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt$. For $x(t) = t^3$ and $y(t) = t^2$, $dx/dt = 3t^2$ and $dy/dt = 2t$. The integral becomes $\int_{0}^{2} \sqrt{(3t^2)^2 + (2t)^2} \, dt = \int_{0}^{2} \sqrt{9t^4 + 4t^2} \, dt$. Solving this integral gives the length as $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$.

The Length of the Curve x(t)=t3,y(t)=t2x ( t ) = t ^ { 3 } , y ( t ) = t ^ { 2 }x(t)=t3,y(t)=t2

The Length of the Curve $x ( t ) = t ^ { 3 } , y ( t ) = t ^ { 2 }$