the area enclosed by y=e^-x^2, the X- axis and the maximum ordinate

Answer 1

You probably meant

y=e^(-x^2)

look at the graph
http://www.wolframalpha.com/input/?i=y%3De%5E(-x%5E2)

the maximum ordinate is y = 1
however, the function never reaches the x-axis, so there is a "gap".
Unless you left out some vertical boundaries, the question is bogus.

Answer 2

maybe not bogus, but not elementary either:

https://en.wikipedia.org/wiki/Gaussian_integral