Chapter 5.3, Problem 180E

The double improper integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\left(-x^2+y^{2/2}\right)dxdy}}}$may be defined as the limit value of the double integrals ${\displaystyle \underset{D_a}{\iint }{e}^{\left(-x^2+y^2/2\right)}dA}$over disks $D_a$of radii a centered at the origin, as a increases without bound; that is,    ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\left(-x^2\right)+y^2/2}dydx{=}_{a\to \infty }^{\lim }}{\displaystyle \underset{D_a}{\iint }{e}^{\left(-x^2+y^{2/2}\right)dA}}}$a. Use polar coordinates to show that ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\left(-x^2+y^{2/2}\right)dydx=2\pi }}}$b. Show that ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{-x^{2/2}dx=\sqrt{2\pi }}}$by using the relation emsp;  ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{\left(-x^2\right)+y^2/2}dydx{=}_{a\to \infty }^{\lim }}{\displaystyle \underset{D_a}{\iint }{e}^{\left(-x^2+y^{2/2}\right)dA}}}$