Find the mass of a thin wire in the shape of a helix given by $\overrightarrow {r}(t)=\langle \cos t,\sin t, t\rangle$ for $0\le t\le6\pi$ with density of $z$ at the point $(x,y,z)$ on the wire (curve)?
Can you all provide me with a hint as to how I should get started with this one?
I know that:
$$M=\int^{x_1}_{x_0}\int^{y_1}_{y_0}\int^{z_1}_{z_0}\sigma(x,y,z)\;dz\;dy\;dx$$
Does this mean that:
$$M=\int^{6\pi}_{0}\int^{6\pi}_{0}\int^{6\pi}_{0}z\;dz\;dy\;dx$$
Best Answer
Let $\mathbf r(t)=(x(t),y(t),z(t))$ .
As @littleO says, you have to evaluate a line integral, that is $$\int_C \sigma(x,y,z)\,ds=\int_a^b\sigma(x(t),y(t),z(t))\, \sqrt {[x'(t)]^2+[y'(t)]^2+[z'(t)]^2}\,dt$$ See my comment also.