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 $\int_{-2}^{3}(x + 1)dx$

(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:

The region is bounded by:

• Line: $y - 1 = x$ which gives $y = x + 1$
• Left boundary: $x = -2$
• Right boundary: $x = 3$
• Bottom: $x$-axis where $y = 0$

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To find where the line crosses the $x$-axis, set $y = 0$:

$0 = x + 1$

$x = -1$

The line crosses the $x$-axis at $x = -1$.

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Checking values at key points:

At $x = -2$: $y = -2 + 1 = -1$ (below $x$-axis)

At $x = -1$: $y = -1 + 1 = 0$ (on $x$-axis)

At $x = 3$: $y = 3 + 1 = 4$ (above $x$-axis)

From $x = -2$ to $x = -1$, the line is below the $x$-axis.

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Checking Statement (A): Area $= \int_{-2}^{3}(x + 1)dx$

This is incorrect because when the curve is below the $x$-axis, the integral gives a negative value, but area must be positive.

The correct formula is:

Area $= \int_{-2}^{-1}|x + 1|dx + \int_{-1}^{3}(x + 1)dx$

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

Statement (A) is incorrect.

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Area below $x$-axis (from $x = -2$ to $x = -1$):

$\int_{-2}^{-1}-(x + 1)dx$

$= -\left[\frac{x^2}{2} + x\right]_{-2}^{-1}$

At $x = -1$: $\frac{(-1)^2}{2} + (-1) = \frac{1}{2} - 1 = -\frac{1}{2}$

At $x = -2$: $\frac{(-2)^2}{2} + (-2) = 2 - 2 = 0$

$= -\left[-\frac{1}{2} - 0\right] = \frac{1}{2}$

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Area above $x$-axis (from $x = -1$ to $x = 3$):

$\int_{-1}^{3}(x + 1)dx$

$= \left[\frac{x^2}{2} + x\right]_{-1}^{3}$

At $x = 3$: $\frac{9}{2} + 3 = \frac{15}{2}$

At $x = -1$: $\frac{1}{2} - 1 = -\frac{1}{2}$

$= \frac{15}{2} - \left(-\frac{1}{2}\right)$

$= \frac{15}{2} + \frac{1}{2}$

$= \frac{16}{2}$

$= 8$

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Total Area:

Area $= \frac{1}{2} + 8$

Area $= \frac{1}{2} + \frac{16}{2}$

Area $= \frac{17}{2}$ sq. units

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Checking all statements:

(A) Area $= \int_{-2}^{3}(x + 1)dx$ is incorrect

(B) Area $= \frac{15}{2}$ sq. units is incorrect

(C) Area $= 8$ sq. units is incorrect

(D) Area $= \frac{17}{2}$ sq. units is correct

Therefore, only statement (D) is correct.