Section 9.2: 7, 9, 10

Section 9.4: 7, 10

And: Let $\displaystyle{I = \int_{-\infty}^\infty e^{-\frac{x^2}{2}}dx}$.

1. Show that $\displaystyle{I^2 = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-\frac{x^2 + y^2}{2}}dxdy}$.
2. Use polar coordinates to show that $I^2 = 2\pi$ and, hence, that $I = \sqrt{2\pi}$.