Brain Gains (Rapid Recall, Jivin' Generation)

  1. Set up a triple integral that would give the volume of the solid region in space that is bounded by the surfaces $z = 2-y^2$, $y+z=0$, $x=0$ and $x=3$. Construct a rough sketch of the region as part of your work.
  2. Draw the region whose area is given by $\ds\int_0^{\pi/2}\int_0^{3\sin(2\theta)}rdrd\theta$.
  3. Set up an iterated integral formula that would give the $x$-coordinate of the centroid of the right most petal of the rose $r = 5\cos(2\theta)$.
  4. Convert the integral $\ds\int_0^{3}\int_0^{x}xydydx$ to polar coordinates.
  5. A wire is parametrized by $\vec r(t) = (2\cos t, t^2, 3t)$ for $0\leq t\leq 7$. The temperature at points on the wire is given by $T(x,y,z)=x^2+yz$. Set up an iterated integral formula that would give the average temperature of the wire.

Group Problems

  1. Set up a triple integral that would give the volume of the solid region that is bounded above by the paraboloid $z=6-x^2-y^2$, and below by the cone $z = \sqrt{x^2+y^2}$. Construct a rough sketch of the region as part of your work.
  2. Set up a triple integral that would give the volume of the solid region in the first octant that is below the plane $z=9$ and above the paraboloid $z=x^2+y^2$.
  3. A metal plate lies over the right most petal of the rose $r = 4\sin(3\theta)$. The density at points on the platen is give by $\delta(x,y) = xy^2$. Set up a double integral in polar coordinates to give the average density.
  4. A metal plate lies over one the right most petal of the rose $r = 3\cos(2\theta)$. The density at points on the plate is given by $\delta(x,y) = x$. Set up a double integral in polar coordinates to give the $y$-coordinate of the center of mass.