Rapid Recall
- Give the $(x,y,z)$ coordinates of the point in space given by $(\rho,\phi,\theta) = (2,\pi/2,\pi)$.
Solution
- A tennis ball has been cut in half. Suppose the inner radius of the tennis ball is 3 cm, and the outer radius is 3.5 cm. Give bounds of the form $?\leq \theta\leq ?$, $?\leq \phi\leq ?$, $?\leq \rho\leq ?$, to describe the bottom half ($z\leq 0$) of the tennis ball.
Solution
- Consider the sphere $x^2+y^2+z^2=9$. Give a cylindrical equation for the sphere. Then give a spherical equation for the sphere.
Solution
Group problems
- Set up an integral formula to compute each of the following:
- The average temperature of a metal plate in the $xy$-plane bounded by the curves $y=8-x^2$ and $y=x+2$, where the temperature at points on the plate is given by $f(x,y)=x+y^2$.
- The average charge density on a wire that lies along the helix $\vec r(t) = (3\cos t,3\sin t, 4t)$ for $0\leq t\leq 4\pi$, provided the charge at each point on the wire is given by $\sigma(x,y,z) = x^2+y^2+z$.
- The average pressure in a solid region in space inside the sphere $x^2+y^2+z^2=9$, provided the pressure at each point in the sphere 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 Cartesian coordinates $(x,y,z)$ for the spherical coordinates $(\rho,\phi,\theta)$ given by $(2,\pi/2,\pi)$, $(2,\pi,\pi/2)$, $(2,0,3\pi)$, and $(4,\pi/4,\pi/2)$.
- Give spherical coordinates $(\rho,\phi,\theta)$ for the Cartesian coordinates $(x,y,z)$ given by $(0,3,0)$, $(0,0,3)$, and $(1,2,2)$, $(3,0,4)$.
- 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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