Find the mass of a wire formed by the intersection of the sphere $$ x^2 + y^2 + z^2 = 1 $$and the plane $$ x + y + z = 0 $$if the density at $$(x, y, z)$$ is given by $$p(x, y, z) = x^2$$ per unit length of wire.
I really have no idea how to start this question. Can anyone give me some sort of hints? :o
One way to parameterise your curve would be $$\mathbf{x}(t) = \frac{1}{\sqrt{2}} (0,1, -1) \cos t + \frac{1}{\sqrt{6}}(-2, 1, 1) \sin t$$
Then solve for $$\int_0^{2 \pi} \rho(\mathbf{x}(t)) \|\dot{\mathbf{x}}(t)\| dt $$
Note that $\|\dot{\mathbf{x}}(t)\| = 1$ by the choice of parametrisation and so
$$\text{mass} = \int_0^{2 \pi} \frac{2}{3} \sin^2t \hspace{0.1cm} dt = \frac{2 \pi}{3} $$
I should explain how that parameterisation works. I know any plane through the origin intersects the unit sphere along a great circle, and so I can find two orthogonal vectors both on the plane and the sphere and use $\sin(t)$ and $\cos(t)$ to generate the whole circle.
