Rapid Recall
- Draw the solid whose volume is given by the integral $\ds\int_{0}^{\pi}\int_{\pi/3}^{\pi/2}\int_{0}^{3}\rho^2\sin\phi d\rho d\phi d\theta$.
Solution
I'll draw this on the board. It looks somewhat like a circular bunt cake.
- Set up an integral formula to compute the $z$-coordinate of the center of mass of the solid above, provided the density is given by $\delta = x^2+y^2+z^2$.
Solution
The bounds don't change at all, rather we just have to add the correct pieces from the center-of-mass formulas. Note that $z=\rho\cos\phi$ and $\delta = \rho^2$. $$\bar z = \frac{\iiint_Dz\delta dV}{\iiint_D\delta dV}=\frac{\ds\int_{0}^{\pi}\int_{\pi/3}^{\pi/2}\int_{0}^{3}(\rho \cos\phi)(\rho^2) \rho^2\sin\phi d\rho d\phi d\theta}{\ds\int_{0}^{\pi}\int_{\pi/3}^{\pi/2}\int_{0}^{3}(\rho^2) \rho^2\sin\phi d\rho d\phi d\theta}$$
- The temperature at points in the solid above is given by $T(x,y,z) = x+2y$. Set up an integral that would find the average temperature in the solid.
Solution
This is almost exactly the same as above. There is no density, and instead of finding the average $z$ value, we now want the average $T$ value. So just swap out $z$ for $T$ and we get $$\bar T = \frac{\iiint_D T dV}{\iiint_D dV}=\frac{\ds\int_{0}^{\pi}\int_{\pi/3}^{\pi/2}\int_{0}^{3}(\rho\sin\phi\cos\theta+2\rho\sin\phi\sin\theta) \rho^2\sin\phi d\rho d\phi d\theta}{\ds\int_{0}^{\pi}\int_{\pi/3}^{\pi/2}\int_{0}^{3} \rho^2\sin\phi d\rho d\phi d\theta}$$
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 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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