Brain Gains (Rapid Recall, Jivin' Generation)
- Set up an iterated triple integral that gives the volume of the solid in the first octant that is bounded by the coordinate planes ($x=0$, $y=0$, $z=0$), the plane $y+z=2$, and the surface $x=4-y^2$, using the order of integration $dxdzdy$. Make sure you sketch the region.
- Set up an integral to give the volume of the pyramid in the first octant that is below the planes $\ds\frac{x}{3}+\frac{z}{2}=1$ and $\ds\frac{y}{5}+\frac{z}{2}=1$. [Hint, don't let $z$ be the inside bound. Try an order such as $dydxdz$.]
- A wire lies along the quarter circle $\vec r(t) = (a \cos t, a\sin t)$ for $0\leq t\leq \pi/2$. Set up an integral that would compute the $x$-coordinate of the center-of-mass of the wire.
- A metal plate lies inside the circle $x^2+y^2=a^2$ in the first quadrant. Set up an integral that would compute the $x$-coordinate of the center-of-mass of the metal plate.
Group Problems
- 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 $V=\frac{4}{3}\pi 5^3$.
- 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.
- 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.)