Answer
The area is 1.
Work Step by Step
We have
$x\left( t \right) = {{\rm{e}}^t}$, $x'\left( t \right) = {{\rm{e}}^t}$,
$y\left( t \right)=t$.
Using Eq. (11) the area is
$A = \mathop \smallint \limits_{t = 0}^1 t\left( {{{\rm{e}}^t}} \right){\rm{d}}t = \mathop \smallint \limits_{t = 0}^1 t{{\rm{e}}^t}{\rm{d}}t$.
Note that the indefinite form of this integral has been evaluated in Example 2 of Section 8.1.
So, using integration by parts, it follows that
$A = \mathop \smallint \limits_{t = 0}^1 t\left( {{{\rm{e}}^t}} \right){\rm{d}}t = \mathop \smallint \limits_{t = 0}^1 t{{\rm{e}}^t}{\rm{d}}t = t{{\rm{e}}^t}|_0^1 - {{\rm{e}}^t}|_0^1 = {\rm{e}} - {\rm{e}} + 1$.
$A = 1$.
