Answer
\[A=12\]
Work Step by Step
\[\begin{align}
& \text{From the graph} \\
& R=\left\{ \left( x,y \right):x\le y\le 2x+1,\text{ }0\le x\le 4\text{ } \right\} \\
& \text{Then,} \\
& A=\int_{0}^{4}{\int_{x}^{2x+1}{dy}dx} \\
& \text{Integrate} \\
& A=\int_{0}^{4}{\left[ y \right]_{x}^{2x+1}dx} \\
& A=\int_{0}^{4}{\left( 2x+1-x \right)dx} \\
& A=\int_{0}^{4}{\left( x+1 \right)dx} \\
& A=\left[ \frac{1}{2}{{x}^{2}}+x \right]_{0}^{4} \\
& A=\frac{1}{2}{{\left( 4 \right)}^{2}}+\left( 4 \right) \\
& A=12 \\
\end{align}\]
