Calculate the arc length of the indicated portion of the curve r(t).
-$$r ( t ) = 2 t ^ { 4 } \mathbf { i } + 10 t ^ { 4 } \mathbf { j } + 11 t ^ { 4 } k _ { , } - 1 \leq t \leq 3$$
A) 300
B) 4500
C) 18,000
D) 1200

Question

Quizplus · Accepted Answer

To find the arc length, we use the formula 
$$L=\int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2}dt$$
where $a$ and $b$ are the lower and upper limits of the parameter $t$. 
In this case, we have 
$$L=\int_{-1}^{3}\sqrt{\left(\frac{d}{dt}\left(2t^4\right)\right)^2+\left(\frac{d}{dt}\left(10t^4\right)\right)^2+\left(\frac{d}{dt}\left(11t^4\right)\right)^2}dt$$
$$=\int_{-1}^{3}\sqrt{16t^6+400t^6+484t^6}dt$$
$$=\int_{-1}^{3}\sqrt{900t^6}dt$$
$$=30\int_{-1}^{3}t^3dt$$
$$=\left[\frac{30}{4}t^4\right]_{-1}^{3}$$
$$=30\cdot\frac{3^4-(-1)^4}{4}$$
$$=\boxed{1200}$$

Calculate the Arc Length of the Indicated Portion of the Curve