- Let $P=(3,0,0)$, $Q=(0,2,0)$, and $R=(0,0,1)$.
- Find a vector that is orthogonal to both $\vec{PQ}$ and $\vec {PR}$. Call it $\vec n$.
- Find the area of the triangle $\Delta PQR$.
- Let $S=(x,y,z)$ be any point on the plane PQR. What is $\vec {PS}\cdot \vec n$ (you should get a number)? Then compute and expand the dot product, to get an equation of the tangent plane.
- Set up an integral formula to compute each of the following:
- The mass of a disc that lies inside the circle $x^2+y^2=9$ and has density function given by $\delta = x+10$
- The $x$-coordinate of the center of mass (so $\bar x$) of the disc above.
- The $z$-coordinate of the center-of-mass (so $\bar z$) of the solid object in the first octant (all variables positive) that lies under the plane $2x+3y+6z=6$ (Draw the region as well).
- The $y$-coordinate of the center-of-mass (so $\bar y$) of the same 3D solid.
- Draw the region described the bounds of each integral.
- $\ds\int_{0}^{3}\int_{0}^{\pi}\int_{0}^{5}rdzdrd\theta$
- $\ds\int_{-1}^{1}\int_{0}^{1-y^2}\int_{0}^{x}dzdxdy$
- A wire lies along the curve $C$ parametrized by $\vec r(t) = (t^2+1, 3t, t^3)$ for $-1\leq t\leq 2$.
- Compute $ds$. (Remember - a little distance equals the product of the speed and a little time.)
- Set up an integral to find $\bar x$, then $\bar y$, then $\bar z$, for the centroid of $C$.
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