Find the exact mass of a thin wire in the shape of the helix $x = 2 \sin ( t ) , y = 2 \cos ( t ) , z = 3 t , 0 \leq t \leq 2 \pi$ if the density is 5
A) $8 \sqrt { 10 } \pi ^ { 2 }$
B) $\sqrt { 13 } \pi$
C) $10 \sqrt { 13 } \pi$
D) $4 \sqrt { 13 }$
E) $10 \sqrt { 13 }$

Question

Quizplus · Accepted Answer

The mass of the wire is found by integrating the density over the length of the wire. The length element $ds$ is given by $\sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2} \, dt$. For the given helix, $dx/dt = 2 \cos(t)$, $dy/dt = -2 \sin(t)$, and $dz/dt = 3$. Thus, $ds = \sqrt{(2 \cos(t))^2 + (-2 \sin(t))^2 + 3^2} \, dt = \sqrt{4 + 9} \, dt = \sqrt{13} \, dt$. The mass is $\int_0^{2\pi} 5 \sqrt{13} \, dt = 10 \sqrt{13} \pi$.

Find the Exact Mass of a Thin Wire in the Shape