- Set up an integral formula to compute the $z$-coordinate of the center-of-mass (so $\bar z$) of the solid object in the first octant (all variables positive) that lies inside the sphere $x^2+y^2+z^2=9$.
- Let $P=(1,2,0)$, $Q=(0,2,-1)$, and $R=(3,0,4)$.
- Find a vector that is orthogonal to both $\vec{PQ}$ and $\vec {PR}$.
- Find the area of triangle $\Delta PQR$.
- Give an equation of the plane PQR. (Let $S=(x,y,z)$ be any point on the plane PQR. Use $\vec {PS}\cdot (\vec {PQ}\times \vec {PR})=0$.)
- Let $T$ be another point in space. The quantity $\left|\text{proj}_{\vec {PQ}\times \vec {PR}}\vec {PT}\right|$ computes the distance between two things. What two things?
- Draw the region described the bounds of each integral.
- $\ds\int_{0}^{3}\int_{0}^{\pi}\int_{0}^{5}rdzd\theta dr$
- $\ds\int_{-1}^{1}\int_{0}^{1-y^2}\int_{0}^{x}dzdxdy$
- $\ds\int_{0}^{2}\int_{0}^{1-y/2}\int_{6+z}^{6-z}dxdzdy$
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