Question

A spring has been formed from wire with a variable density into the shape of helix, parameterized by r(t)=(2cost, 2sint, t) for 0≤t≤2pi. Suppose the density of the wire at the point (x, y, z) be given by f(x,y)= 1+y^2+xz (in g/cm). FInd the mass of the spring.

Answer 1

The mass will be given by
\[\large \begin{align*}M&=\int_C f({\bf r}(t))~d{\bf r}\\\\
&=\int_0^{2\pi} f(2\cos t,2\sin t,t)\|{\bf r}'(t)\|~dt\\\\
&=\int_0^{2\pi} (1+4\sin^2t+2t\cos t)\|\langle -2\sin t,2\cos t,1\rangle\|~dt
\end{align*}\]

Answer 2

thank you so much.