$$\text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$$
-$$r ( t ) = ( 3 \cos t ) \mathbf { i } + ( 3 \sin t ) \mathbf { j } + 3 t k , 0 \leq t \leq 2 \pi ; \delta = 5$$
A) $30 \pi \sqrt { 2 }$ units
B) $6 \pi \sqrt { 2 }$ units
C) $90 \pi \sqrt { 2 }$ units
D) $10 \pi$ units

Question

Quizplus · Accepted Answer

The mass of the wire can be calculated using the formula $m = \int_{a}^{b} \delta \, ds$, where $ds$ is the arc length differential and $\delta$ is the density of the wire. First, we find the derivative of $r(t)$ with respect to $t$: $$r'(t) = (-3\sin t) \mathbf{i} + (3\cos t) \mathbf{j} + 3\mathbf{k}.$$ The magnitude of $r'(t)$ is $$\sqrt{(-3\sin t)^2 + (3\cos t)^2 + 3^2} = \sqrt{9\sin^2 t + 9\cos^2 t + 9} = \sqrt{18} = 3\sqrt{2}.$$ Thus, $ds = 3\sqrt{2} dt$. The mass is then $$m = \int_{0}^{2\pi} 5 \cdot 3\sqrt{2} dt = 15\sqrt{2} \int_{0}^{2\pi} dt = 15\sqrt{2} \cdot 2\pi = 30\pi\sqrt{2}.$$

Find the mass of the wire that lies along the curve r and has density δ. \text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. } Find the mass of the wire that lies along the curve r and has density δ.

$\text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$