Consider the region bounded by the lines $y - 1 = x, x = -2, x = 3$ and x-axis. Then

(A) The area of the bounded region is given by $\int\limits_{-2}^3(x + 1)dx$
(B) The numerical value of the area is $\frac{15}{2}$ sq. units
(C) The numerical value of the area is 8 sq. units
(D) The numerical value of the area is $\frac{17}{2}$ sq. units

Choose the correct answer from the options given below:

(D) only

The correct answer is Option (4) → (D) only

$\text{Line: } y = x+1$

$\text{Intersects x–axis when } x+1 = 0 \Rightarrow x = -1$

$\text{Region is between } x=-2,\; x=-1,\; x=3$

$A = \int_{-2}^{-1} -(x+1)\,dx \;+\; \int_{-1}^{3} (x+1)\,dx$

$A_1 = \int_{-2}^{-1} -(x+1)\,dx = \left[-\frac{x^2}{2} - x\right]_{-2}^{-1} = \frac12$

$A_2 = \int_{-1}^{3} (x+1)\,dx = \left[\frac{x^2}{2}+x\right]_{-1}^{3} = 8$

$A = A_1 + A_2 = \frac12 + 8 = \frac{17}{2}$