Question

The area between the curves y = xe^x and y = xe^–x and the line x = 1 is

• A. 2e
• B. e
• C. 2/e
• D. 1/e

Answer 1

Answer: (C)

2/e

Answer 2

The line x = 1 meets the curves in A(1, e) and B(1, 1/e). Both the curves pass through the origin.

The required area

A =$\int _ { 0 } ^ { 1 } \left( x e ^ { x } - x e ^ { - x } \right) d x$

= $\left[ x \left\{ e ^ { x } + e ^ { - x } \right\} \right] _ { 0 } ^ { 1 } - \int _ { 0 } ^ { 1 } \left( e ^ { x } + e ^ { - x } \right) \cdot 1 d x$

= $\left( e + \frac { 1 } { e } \right) - \left[ e ^ { x } - e ^ { - x } \right] _ { 0 } ^ { 1 }$=$\left( e + \frac { 1 } { e } \right) - \left( e - \frac { 1 } { e } \right)$

= 2/e sq. units