Answer
4.644
Work Step by Step
First step is to find the point of intersection of curves $y=e^{x},y=e^{3x}$ and $x=1$ .
For this,
$e^{x}=e^{3x}$
$x=3x$
$2x=0$
$x=0$
Thus, the point of intersection for given curves is (0, 1).
Let A be the area of the region bounded by the curves, which is calculated as follows:
$A=\int_ {0}^{1}(e^{3x}-e^{x})dx$
$=\int_ {0}^{1}e^{3x}dx-\int_ {0}^{1}e^{x}dx$
$=[\frac{1}{3}e^{3x}-e^{x}]_{0}^{1}$
$\approx 4.644$
Hence, the area of the region bounded by the curves is A =4.644 .
