The area of the region bounded by the parabola $(y-2)^{2}=x-1,$ the tangent to the parabola at the point (2, 3) and the x-axis is:

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Accepted Answer

To determine the area of the region bounded by the parabola $(y-2)^{2} = x-1$, the tangent to the parabola at the point (2, 3), and the x-axis, we need to follow these steps:

1. **Equation of the Parabola**: The given equation of the parabola is $(y-2)^{2} = x-1$. This can be rewritten as $x = (y-2)^{2} + 1$.

2. **Finding the Tangent Line**: To find the equation of the tangent line at the point (2, 3), we first need the derivative of $x = (y-2)^{2} + 1$ with respect to $y$. Differentiating, we get:
   $$
   \frac{dx}{dy} = 2(y-2)
   $$
   At the point (2, 3), $y = 3$, so:
   $$
   \frac{dx}{dy} = 2(3-2) = 2
   $$
   The slope of the tangent line is 2. Using the point-slope form of a line, the equation of the tangent line at (2, 3) is:
   $$
   x - 2 = 2(y - 3)
   $$
   Simplifying, we get:
   $$
   x = 2y - 4
   $$

3. **Finding Points of Intersection**: We need to find the points where the tangent line intersects the parabola and the x-axis. Setting the equations equal:
   $$
   (y-2)^{2} + 1 = 2y - 4
   $$
   Simplifying, we get:
   $$
   (y-2)^{2} = 2y - 5
   $$
   Solving this quadratic equation, we find the points of intersection. However, since the tangent line is at (2, 3), it touches the parabola at this point and intersects the x-axis at $y = 0$.

4. **Area Calculation**: The region bounded by the parabola, the tangent line, and the x-axis is a vertical strip from $y = 0$ to $y = 3$. The area can be calculated by integrating the difference between the parabola and the tangent line from $y = 0$ to $y = 3$:
   $$
   \text{Area} = \int_{0}^{3} \left((y-2)^{2} + 1 - (2y - 4)\right) \, dy
   $$
   Simplifying the integrand:
   $$
   (y-2)^{2} + 1 - 2y + 4 = y^{2} - 4y + 4 + 1 - 2y + 4 = y^{2} - 6y + 9
   $$
   Integrating:
   $$
   \int_{0}^{3} (y^{2} - 6y + 9) \, dy = \left[\frac{y^{3}}{3} - 3y^{2} + 9y\right]_{0}^{3}
   $$
   Evaluating the integral:
   $$
   \left(\frac{27}{3} - 27 + 27\right) - \left(0\right) = 9
   $$

Thus, the area of the region is 9, which confirms that Option 2 is correct.

**Explanation for Incorrect Options**:
- **Option 1 (6)**: This value is incorrect because the integration of the area calculation yields 9, not 6.
- **Option 3 (12)**: This value is incorrect as the calculated area from the integration is 9, not 12.
- **Option 4 (3)**: This value is incorrect because the integration result is 9, not 3.

Therefore, the correct answer is indeed Option 2, which is 9.