Let $0 < a < 1$. Let $S ( a )$ denote the sum of the areas of two regions, one region bounded by the curve $y = x e ^ { 2 x }$, the $x$-axis, and the straight line $x = a - 1$, and the other region bounded by the curve $y = x e ^ { 2 x }$, the $x$-axis, and the straight line $x = a$. We are to find the value of $a$ at which $S ( a )$ is minimized.
The indefinite integral of $x e ^ { 2 x }$ is to be determined, where $C$ is the constant of integration.
The value of $x e ^ { 2 x }$ is $x e ^ { 2 x } < 0$ for $x < 0$ and $x e ^ { 2 x } \geqq 0$ for $x \geqq 0$. Hence we have
$$S ( a ) = \frac { \mathbf { L M } } { \mathbf { N } } \left\{ \mathbf { O } + \left( \mathbf { P } a - \mathbf { Q } \right) e ^ { 2 ( a - 1 ) } + ( \mathbf { R } a - 1 ) e ^ { 2 a } \right\} .$$
Further, since
$$S ^ { \prime } ( a ) = ( a - \mathbf { S } ) e ^ { 2 ( a - 1 ) } + a e ^ { 2 a } ,$$
the value of $a$ at which $S ( a )$ is minimized is $a = \dfrac { \square \mathbf { T } } { e ^ { 2 } + \mathbf { U } }$, which satisfies $0 < a < 1$.