- 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 $\bar z$, the $z$-coordinate of the centroid of the region in space above the $xy$-plane that is bounded above by the plane $z=8$ and below by the cone $z^2=x^2+y^2$.
- 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 $y=x$, $y=2$, and $x=0$, where the temperature at points on the plate is given by $f(x,y)=\sin(y^2)$.
- 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 $\sigma(x,y,z) = xz$.
- 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 $p(x,y,z) = 10+x$.
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