Find the area of the largest rectangle that can be inscribed under the curve y = e^(-x^2) in the first and second quadrants.

To find the area of the largest rectangle that can be inscribed under the curve y = e^(-x^2) in the first and second quadrants, we need to maximize the product of the width and height of the rectangle.

Let's start by visualizing the problem.

The graph of y = e^(-x^2) is a standard bell curve symmetric about the y-axis. We are interested in finding the rectangle that fits snugly under the curve.

Since the curve is symmetric, we will focus on finding the rectangle in the first quadrant and then double the area to account for the second quadrant.

Now, let's proceed with the steps to solve the problem:

Step 1: Determine the x-coordinate for the rectangle's vertices in the first quadrant.
To maximize the area, we need to find the maximum width of the rectangle, which occurs at the x-coordinate where the slope of the curve is zero. We can find this by taking the derivative of y = e^(-x^2) with respect to x and setting it equal to zero.

dy/dx = -2xe^(-x^2) = 0

Solving this equation, we get x = 0.

Step 2: Determine the corresponding y-coordinate for the rectangle's vertices in the first quadrant.
Substituting x = 0 into y = e^(-x^2), we find y = 1.

Therefore, the two vertices in the first quadrant are (0, 0) and (0, 1).

Step 3: Calculate the area of the rectangle in the first quadrant.
The width of the rectangle is twice the x-coordinate of the first quadrant vertices, which is 2x = 2(0) = 0.
The height of the rectangle is the y-coordinate of the first quadrant vertices, which is y = 1.

So, the area of the rectangle in the first quadrant is 0 multiplied by 1, which is 0.

Step 4: Double the area to account for the second quadrant.
Since the first quadrant is symmetrical to the second quadrant, we can double the area calculated in step 3 to get the area of the largest rectangle that can be inscribed.

Doubling 0 gives us an area of 0 in the second quadrant.

Therefore, the area of the largest rectangle that can be inscribed under the curve y = e^(-x^2) in the first and second quadrants is 0.