the area enclosed by y=e^-x^2, the X- axis and the maximum ordinate
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.
maybe not bogus, but not elementary either:
https://en.wikipedia.org/wiki/Gaussian_integral
To find the area enclosed by the curve y = e^(-x^2), the x-axis, and the maximum ordinate, we need to integrate the function and determine the limits of integration.
Step 1: Finding the maximum ordinate
To find the maximum ordinate, we need to locate the peak of the curve. Differentiating the function y = e^(-x^2) with respect to x, we get:
dy/dx = -2x * e^(-x^2)
To find the peak, we set dy/dx = 0 and solve for x:
-2x * e^(-x^2) = 0
This equation implies that either x = 0 or e^(-x^2) = 0. However, e^(-x^2) is never equal to zero, so we only consider the case x = 0.
Therefore, the maximum ordinate occurs at x = 0, and the corresponding y-value is y = e^(-0^2) = e^0 = 1.
Step 2: Determining the limits of integration
The curve y = e^(-x^2) is symmetric about the y-axis. So, the enclosed area is the total area in the first quadrant, which is then doubled to account for the symmetry.
Since the maximum ordinate occurs at x = 0, and the curve goes to zero as x approaches infinity, the limits of integration will be from 0 to positive infinity.
Step 3: Integrating the function
Now, we integrate the function y = e^(-x^2) from 0 to infinity to find the area:
A = 2 * ∫[0, ∞] e^(-x^2) dx
However, the integral of e^(-x^2) with respect to x does not have an elementary antiderivative. Therefore, a standard mathematical function cannot be used to evaluate the integral.
Instead, we need to rely on numerical methods or approximation techniques to find the value of the integral.