# The area of the region bounded by the curve $y=x+1$ and the lines $x=2$ and $x=3$ is $(A)\frac{7}{2} sq.units\quad(B)\;\frac{9}{2}sq. units\quad(C)\frac{11} {2}sq.units\quad(D)\frac{13}{2} sq.units$

Toolbox:
• The area enclosed by the curve $y=f(x)$,the $x$-axis and the ordinates $x=a$ and $x=b$ is given by $\int_a^b ydx.$
• $\int x^ndx=\large\frac{x^{n+1}}{n+1}$$+c$
Step 1:
Given curve is $y=x+1$ which is a straight line bounded between the lines $x=2$ and $x=3$
The required area is the shaded portion as shown in the fig.
Area of the shaded portion is $A=\int_2^3(x+1)dx.$
Step 2:
On integrating we get,
$\begin{bmatrix}\large\frac{x^2}{2}\normalsize +x\end{bmatrix}_2^3$
On applying limits we get,
$\begin{bmatrix}\large\frac{1}{2}\normalsize(3^2-2^2)+(3-2)\end{bmatrix}=\begin{bmatrix}\large\frac{1}{2}\normalsize(9-4)+(3-2)\end{bmatrix}$
$\qquad\qquad\qquad\qquad\qquad=\begin{bmatrix}\large\frac{5}{2}\normalsize +1\end{bmatrix}=\large\frac{7}{2}$sq.units.
Hence the correct option is A.