If S be the area of the region enclosed by y=e^-x^2, y=0,...

If $S$ be the area of the region enclosed by $y=e^{-x^{2}}, y=0, x=0$ and $x=1$. Then, [More than One Option Correct 2012] (a) $S \geq \frac{1}{e}$ (b) $S \geq 1-\frac{1}{e}$ (c) $S \leq \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$ (d) $S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$

If $S$ be the area of the region enclosed by $y=e^{-x^{2}}, y=0, x=0$ and $x=1$. Then,
[More than One Option Correct 2012]
(a) $S \geq \frac{1}{e}$
(b) $S \geq 1-\frac{1}{e}$
(c) $S \leq \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$
(d) $S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$

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Key Concepts

Definite Integration

Definite integration is the process of calculating the accumulation of quantities, such as the area under a curve, over a specific interval. In this context, it is used to compute the area under the curve of the function, which represents the region of interest.

Exponential Decay Functions

Exponential decay functions, like the one given, decrease as the independent variable increases. Their properties, such as having a maximum value at one end of the interval and a lower value at the other, are crucial for understanding how the function behaves over the interval of integration.

Integral Inequalities

Integral inequalities involve estimating or bounding the value of an integral by comparing the function with simpler functions or using known limits. These techniques help in determining upper and lower bounds for the area under the curve, which is essential in solving the problem.

Bounding Techniques

Bounding techniques in integration include methods such as using the maximum and minimum values of the function on the interval to create estimates for the area. This concept is key when evaluating or comparing the area with specific numerical bounds provided in options.

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