Problem: Let $\theta$ represent the usual angle in the $xy$-plane emanating from the origin counter-clockwise from the $x$-axis, and let $\phi$ represent the angle from the $z$-axis to the $xy$-plane. If $\rho$ is distance from the origin, find the volume of the object created by rotating the region where $1\le\rho\le 3$ and $\pi/6\le\phi\le\pi/2$ in a full circle around the $z$-axis.

Solution: In spherical coordinates, we compute as follows.

\[\begin{align*} \int_0^{2\pi}\int_{\pi/6}^{\pi/2}\int_1^3 \rho^2\sin\phi\,d\rho\,d\phi\,d\theta &= \int_0^{2\pi}\,d\theta\int_{\pi/6}^{\pi/2}\sin\phi\,d\phi\int_1^3\rho^2\,d\rho \\ &= 2\pi\left(-\cos\phi\right)\Big|_{\phi=\pi/6}^{\phi=\pi/2}\left(\frac{\rho^3}{3}\right)\Big|_{\rho=1}^{\rho=3}\\ &= 2\pi\cdot\frac{\sqrt{3}}{2}\left(\frac{27}{3}-\frac{1}{3}\right)\\ &= \frac{26\pi\sqrt{3}}{3}\end{align*}\]

Replacement Question: How many states share a border (including corners) with Colorado?

Answer: 7: Kansas, Nebraska, Wyoming, Utah, Arizona, New Mexico, and Oklahoma. There are only two states that share more borders with neighboring states than Colorado does: Tennessee and Missouri both share borders with 8 states (including each other).

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