Brain Gains (Rapid Recall, Jivin' Generation)

  • Set up an iterated triple integral to find the volume of the region in the solid in the first octant that lies below the plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{5}=1$.
  • Set up an iterated integral formula to give the $y$-coordinate of the centroid of the solid pyramid above.
  • The solid pyramid described above has a density given by $\delta = 2xz$ grams per cubic centimeter. Set up the iterated integral formula that would give $\bar z$, the $z$-coordinate of the center-of-mass of the pyramid.
  • Swap the integral $\int_{-3}^{3}\int_{0}^{\sqrt{9-x^2}}e^{-x^2-y^2}dydx$ to polar coordinates.
  • Compute the integral above.
  • A wire is coiled and lies along the path given by $\vec r(t) = (3\cos t, 3\sin t, 4t)$ for $0\leq t\leq 6\pi$. The density is given by $\delta = z$ grams per meter (the wire gets thicker as you move up the helix). Compute the mass of the wire.

Group Problems

  1. Consider the region in the first quadrant that lies below the curve $y=\sqrt{x}$ for $0\leq x\leq 9$
    • Set up iterated double integrals to find the area of the region using the order $dydx$.
    • Set up iterated double integrals to find the area of the region using the order $dxdy$.
  2. Compute the integral $\int_0^3\int_{2x}^6 e^{y^2}dydx$ by first swapping the order of integration (draw the region first).
  3. Set up an iterated triple integral to find the volume inside the sphere $x^2+y^2+z^2=9$. Use software to verify that you get $V=\frac{4}{3}\pi 5^3$.
  4. Draw the 3D solid that lies above the surface $z=\sqrt{x^2+y^2}$ and below the plane $z=3$. Then set up a triple integral formula to compute the $z$ coordinate of the centroid of the object.
  5. Find the center of mass of region in the first quadrant that lies below the parabola $y=ax^2$ and left of the line $x=b$. (This region is called a parabolic spandrel.)