Let `S` be the area of the region enclosed by `y-e^(-x^(2)),y=0, x=0` and `x=1`. Then

Options

A. `S ge (1)/(e)`
B. `S ge -(1)/(e)`
C. `S le (1)/(4)(1+(1)/(sqrt(e)))`
D. `S le (1)/(sqrt(2)) + (1)/(sqrt(e)) (1-(1)/(sqrt(2)))`

Answer

c

Explanation

Clearly , `S=underset(0)overset(1)int e^(-x^(2))dx`

Also, `S gt` area of rectangle OPCR
`implies S gt OP xx OR implies S gt 1 xx(1)/(e) implies S gt (1)/(e)`
Now , `e^(-x^(2))dx ge e^(-x)` for all `x in [0,1]`
`impliesunderset(0)overset(1)e^(-x^(2)) dx gt underset(0)overset(1)e^(-x) dx`
`implies S ge [-e^(-x)]_(0)^(1) implies S ge (-(1)/(e)+1) implies S ge 1-(1)/(e)`
Clearly, `S lt` Area of rectangle OASQ + Area of rectangle PQBU
`implies S le (1)/(sqrt(2))xx1 + (1-(1)/(sqrt(2))) (1)/(sqrt(e)) =(1)/(sqrt(2))+(1)/(sqrt(e)) (1-(1)/(sqrt(2)))`
Since `(1)/(4)(1+(1)/(sqrt(e))) lt 1-(1)/(e) . `Therefore , `S gt (1)/(4) (1+(1)/(sqrt(e)))`.