The area bounded by the curves `y=x e^x ,y=x e^(-x)` and the line `x=1` is `2/e s qdotu n i t s` (b) `1-2/e s qdotu n i t s` `1/e s qdotu n i t s` (d) `1-1/e s qdotu n i t s`

Options

A. `(2)/(e)` sq. units
B. `1-(2)/(e)` sq. units
C. `(1)/(e)` sq. units
D. `1-(1)/(e)` sq. units

Answer

A

Explanation

Curve tracing : `y=x e^(x)`
`"Let "(dy)/(dx)=0rArre^(x)+xe^(x)=0 or x=-1`.
`"Also, at "x=-1,(dy)/(dx)` changes sign from -ve to + ve,
Hence, x=-1 is a point of minima.
When `xrarroo, yraroo`
`"Also "underset(xrarr-oo)limxe^(x)=underset(xrarr-oo)lim(x)/(e^(-x))=underset(xrarr-oo)lim(1)/(e^(-x))=0`
With similar types of arguments, we can draw the graph of `y=x e^(-x)`.

`"Required ara "=int_(0)^(1)xe^(x)dx-int_(0)^(1)xe^(-x)dx`
`=[xe^(x)]_(0)^(1)-int_(0)^(1)e^(x)dx-([-xe^(-x)]_(0)^(1)+int_(0)^(1)e^(-x)dx)`
`=e-(e-1)-(-e^(-1)-(e^(-1)-1))=(2)/(e)` sq. units