the area of the region bounded by the curve $y = e^{2x}$, the $x$-axis, the $y$-axis, and the line $x = 2$…

Explanation:

Step1: Recall the area - under - curve formula

The area $A$ under the curve $y = f(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}f(x)dx$. Here, $y = e^{2x}$, $a = 0$, and $b = 2$. So, $A=\int_{0}^{2}e^{2x}dx$.

Step2: Use substitution for integration

Let $u = 2x$, then $du=2dx$ and $dx=\frac{1}{2}du$. When $x = 0$, $u = 0$; when $x = 2$, $u = 4$. So, $\int_{0}^{2}e^{2x}dx=\frac{1}{2}\int_{0}^{4}e^{u}du$.

Step3: Integrate $e^{u}$

The antiderivative of $e^{u}$ is $e^{u}$. So, $\frac{1}{2}\int_{0}^{4}e^{u}du=\frac{1}{2}[e^{u}]_{0}^{4}$.

Step4: Evaluate the definite - integral

$\frac{1}{2}[e^{u}]_{0}^{4}=\frac{1}{2}(e^{4}-e^{0})=\frac{1}{2}(e^{4}-1)$.

Answer:

$\frac{1}{2}(e^{4}-1)$