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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

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math.stackexchange.com/q/77486/8581 –  B. S. Feb 27 '13 at 7:11
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1 Answer

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.

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