Problem: Find the volume between $z=1-x^2-y^2$ and the $xy$-plane.

Solution: This is an upside-down parabola raised up 1 unit. We could solve this using either of the integrals

\[\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}1-x^2-y^2\,dy\,dx = \int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_0^{1-x^2-y^2}\,dz\,dy\,dx,\]

but that would be a pain. Instead, we use polar coordinates, where $r^2=x^2+y^2$, giving us the same integral in a nicer form:

\[\begin{align*}\int_0^{2\pi}\int_0^1 (1-r^2)r\,dr\,d\theta &= \int_0^{2\pi}\,d\theta\int_0^1 r-r^3\,dr \\ &= 2\pi\left(\frac{r^2}{2}-\frac{r^4}{4}\right)\Big|_{r=0}^{r=1}\\ &= 2\pi\frac{1}{4}\\ &= \frac{\pi}{2}\end{align*}\]

Replacement Question: To within 10%, what is the current population of the United States?

Solution: Various sources put the number between 307 million and 312 million.


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