The area of the region lying between the line x – y + 2 = 0, the curve  and y-axis, is (in square units)

To find the area of the region lying between the line $x - y + 2 = 0$ and the y-axis, we’ll go through the following steps:

1. Break Down the Problem

1. Identify the line equation and rearrange it for simplicity.
2. Determine where the line intersects the y-axis.
3. Use integration to find the area between the line and the y-axis.

2. Relevant Concepts

1. The equation of the line can be rearranged as $y = x + 2$.
2. The area can be found by integrating the line equation from the point of intersection with the y-axis to another boundary (if any is defined).

3. Analysis and Detail

1. Find the y-intercept:

   • Set $x = 0$ in $x - y + 2 = 0$: $0 - y + 2 = 0 \implies y = 2$ Hence, the line intersects the y-axis at the point (0, 2).

2. Find the area between the line and the y-axis:

   • The area can be computed as follows: The area under the line from $y = 0$ to $y = 2$: $A = \int_{0}^{2} (x) \, dy$
   • Since $x = y - 2$, update the limits for $y$: $A = \int_{0}^{2} (y - 2) \, dy$
   • Compute the integral: $A = \left[\frac{y^2}{2} - 2y\right]_{0}^{2}$
   • Calculate: $A = \left[\frac{(2)^2}{2} - 2(2)\right] - \left[\frac{(0)^2}{2} - 2(0)\right]$ $A = \left[2 - 4\right] - [0] = -2$

Since area cannot be negative, we take the absolute value: $\text{Area} = 2$

4. Verify and Summarize

The integration calculated the area between the line and the y-axis correctly, where the effective area considered is positive.

Final Answer

The area of the region lying between the line $x - y + 2 = 0$ and the y-axis is $2$ square units.