Show that the area between … and … under the bell-shaped curve…

Show that the area between $\mu$ and $\mu +\sigma$ under the bell-shaped curve is a fixed number (near 1/3), by substituting $y=$ ______:
$${\int }_{\mu }^{\mu +\sigma }\frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-\mu {)}^2/2{\sigma }^2}dx={\int }_0^1\frac{1}{\sqrt{2\pi }}{e}^{-y^2/2}dy.$$
What is the area between $\mu -\sigma$ and $\mu$? The area outside $(\mu -\sigma ,\mu +\sigma )$?

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §8.4 Probability and Calculus, #21