Multiple integrals and vector calculus (PS3): Surface and flux integrals

2. Mass, center of mass, and moments

A solid hemisphere of uniform density $k$ occupies the volume $x^2 + y^2 + z^2 \leq a^2$ and $z \geq 0$. Using symmetry arguments wherever possible, find (i) its total mass $M$, (ii) the position of its center of mass, and (iii) its moments and products of inertia $I_{xx}$, $I_{yy}$, $I_{zz}$, $I_{xy}$, $I_{yz}$, $I_{zx}$.

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4. Simple flux integrals

If $\mathbf{n}$ is the unit normal to the surface $S$, evaluate $\int \mathbf{r} \cdot \mathbf{n} \, \de{S}$ over a) the unit cube bounded by the coordinate planes and the planes $x =1 $, $y = 1$, and $z = 1$; b) the surface of a sphere of radius $a$ centered at the origin.

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5. Three flux integrals

Evaluate $\int \mathbf{A} \cdot \mathbf{n} \, \de{S}$ for $\mathbf{A} = [y, 2x, -z]$ and $S$ is the surface of the plane $2x + y = 6$ in the first octant cut off by the plane $z = 4$.

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Evaluate $\int \mathbf{A} \cdot \mathbf{n} \, \de{S}$ for $\mathbf{A} = [x+y^2, -2x, 2yz]$ and $S$ is the surface of the plane $2x + y + 2z = 6$ in the first octant.

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Evaluate $\int \mathbf{A} \cdot \mathbf{n} \, \de{S}$ for $\mathbf{A} = [6z, 2x + y, -x]$ and $S$ is the entire surface of the region bounded by the cylinder $x^2 + z^2 = 9$, $x = 0$, $y = 0$, $z = 0$, and $y = 8$.

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