Rapid Recall
- Draw the solid whose volume is given by the integral $\ds\int_{\pi}^{2\pi}\int_{0}^{\pi/4}\int_{2}^{5}\rho^2\sin\phi d\rho d\phi d\theta$.
Solution
I'll draw this on the board. It looks like a snow cone that has been sliced in half vertically, and someone bit the bottom off.
- Find the volume of a parallelepiped whose edges are the vectors $(3,1,2)$, $(-2,0,4)$, and $(-1,-3,5)$.
Solution
We use the triple product. It doesn't matter which order we use, as we'll compute the absolute value at the end. We compute $$(3,1,2)\times (-2,0,4) = (4,-16,2).$$ We then compute $$|(4,-16,2)\cdot (-1,-3,5)| = |-4+48+10| = 54.$$
- Give an equation of a plane that has normal vector $\vec n = (4,5,6)$ and passes through the point $(1,2,3)$.
Solution
An equation of this plane is $$4(x-1)+5(y-2)+6(z-3)=0.$$
Group problems
Test Postponed. The main reason - having it this weekend seems like too much of a rush. There is no need to rush you. Take the quiz. Learn from your mistakes.
- The test will open Friday, and close Tuesday.
- Thursday and Friday this week there is no new preparation. We will be working in small groups in class, using the ideas we've learned so far to explore the final topics of the semester.
- Monday is our review/lesson plan day. The instructions are in the problem set (at the end of chapter 5). Take all quizzes twice by that day, as I will be locking them all after that point.
- Tuesday the exam closes. Remember that the testing center closes during devotional, so plan accordingly.
- Consider the solid region in space below the paraboloid $z=9-x^2-y^2$ and above the $xy$-plane.
- Set up an integral using cylindrical coordinates to find the volume using the order $d\theta dz dr$. Compute the two inside integrals, till only one integral is left.
- Set up an integral using cylindrical coordinates to find the volume using the order $d\theta dr dz$. Compute the two inside integrals, till only one integral is left.
- One of the integrals above we call the shell method, the other we call the disc method. Which is which, and why?
- Set up an integral formula to compute each of the following average values:
- The average temperature of a metal plate in the $xy$-plane bounded by the curves $y=x$, $y=2$, and $x=0$, where the temperature at points on the plate is given by $T(x,y)=\sin(y^2)$.
- The average charge density on a wire that lies along the helix $\vec r(t) = (a\cos t,a\sin t, bt)$ for $0\leq t\leq 4\pi$, provided the charge at each point on the wire is given by $\rho(x,y,z) = xz$.
- The average pressure in a solid region in space that lies above the cone $z^2=4x^2+4y^2$ and below the paraboloid $z=9-x^2-y^2$, provided the pressure at each point is given by $p(x,y,z) = 10+x$.
- The spherical change-of-coordinates is given by $$(x,y,z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi).$$
- Give an equation of the sphere $x^2+y^2+z^2=9$ in spherical coordinates.
- Give an equation of the cone $x^2+y^2=z^2$ in spherical coordinates.
- Set up an integral to find the volume of the region in space above the $xy$-plane that is bounded above by the sphere $x^2+y^2+z^2=9$ and below by the cone $z^2=x^2+y^2$. The Jacobian for spherical coordinates is $|\rho^2\sin\phi|$.
- Give an equation of the plane $z=8$ in spherical coordinates.
- Set up an integral to find the volume of the region in space above the $xy$-plane that is bounded above by the plane $z=8$ and below by the cone $z^2=x^2+y^2$.
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