Key Concepts and Formulas

• Equation of a Tangent Line: For a curve $y = f(x)$, the equation of the tangent line at the point $(x_1, y_1)$ is given by $y - y_1 = f'(x_1)(x - x_1)$, where $f'(x_1)$ is the derivative of $f(x)$ evaluated at $x_1$, representing the slope of the tangent at that point.
• Area Between Two Curves: If a region is bounded by two continuous curves $y = f(x)$ and $y = g(x)$ between $x = a$ and $x = b$, and $f(x) \ge g(x)$ for all $x \in [a, b]$, then the area $A$ is given by $A = \int_a^b (f(x) - g(x)) dx$, where $f(x)$ is the upper curve and $g(x)$ is the lower curve.

Step-by-Step Solution

Step 1: Find the equation of the tangent line to the parabola at (2, 3)

• What and Why: We need to find the equation of the tangent line because it is one of the boundaries of the region whose area we want to calculate. We use the point-slope form of a line.
• Calculations: The given parabola is $y = f(x) = x^2 - 1$. The derivative of $f(x)$ is: $f'(x) = \frac{d}{dx}(x^2 - 1) = 2x$ At the point (2, 3), the slope of the tangent is: $m = f'(2) = 2(2) = 4$ Using the point-slope form $y - y_1 = m(x - x_1)$ with $(x_1, y_1) = (2, 3)$ and $m = 4$: $y - 3 = 4(x - 2)$ $y - 3 = 4x - 8$ $y = 4x - 5$
• Explanation: The equation of the tangent line is $y = 4x - 5$. We'll call this $y_T(x) = 4x - 5$.

Step 2: Determine the boundaries and limits of integration

• What and Why: We need to define the region bounded by the given curves. The problem states that the area is bounded by the parabola $y = x^2 - 1$, the tangent line $y = 4x - 5$, and the y-axis ($x=0$).
• Calculations:
  1. Parabola: $y_P(x) = x^2 - 1$
  2. Tangent line: $y_T(x) = 4x - 5$
  3. y-axis: $x = 0$ The point of tangency is at $x = 2$. The region is bounded by the y-axis ($x = 0$) on one side and the point of tangency ($x = 2$) on the other. Thus, the limits of integration are $a = 0$ and $b = 2$.
• Explanation: The parabola opens upwards with a vertex at (0, -1). The tangent line passes through (2, 3). Since the parabola opens upwards, the tangent line will always be below the parabola in the region of interest.

Step 3: Set up the definite integral

• What and Why: Now that we have the upper and lower curves and the limits of integration, we can set up the integral to calculate the area.
• Calculations: The area $A$ is given by: $A = \int_a^b (y_{upper} - y_{lower}) dx$ Since $y_P(x) \ge y_T(x)$ in the interval [0, 2], we have: $A = \int_0^2 (y_P(x) - y_T(x)) dx$ $A = \int_0^2 ((x^2 - 1) - (4x - 5)) dx$ $A = \int_0^2 (x^2 - 4x + 4) dx$
• Explanation: We subtract the equation of the lower curve (tangent) from the upper curve (parabola) and integrate over the interval [0, 2].

Step 4: Evaluate the definite integral

• What and Why: We perform the integration and evaluate it at the limits to find the numerical value of the area.
• Calculations: $A = \int_0^2 (x^2 - 4x + 4) dx$ $A = \left[ \frac{x^3}{3} - 2x^2 + 4x \right]_0^2$ $A = \left( \frac{2^3}{3} - 2(2^2) + 4(2) \right) - \left( \frac{0^3}{3} - 2(0^2) + 4(0) \right)$ $A = \left( \frac{8}{3} - 8 + 8 \right) - (0)$ $A = \frac{8}{3}$
• Explanation: The result of the integration is $\frac{8}{3}$, which represents the area of the region in square units.

Common Mistakes & Tips to Avoid

1. Sketching the Region: Always sketch the region to correctly identify the limits of integration and the upper and lower curves.
2. Algebraic Errors: Pay close attention to signs and algebraic manipulations, especially when simplifying the integrand and evaluating at the limits.
3. Correctly Identifying Upper and Lower Curves: Ensure that you are subtracting the lower curve from the upper curve.

Summary

We found the area bounded by the parabola $y = x^2 - 1$, the tangent line at the point (2, 3), and the y-axis. We first found the equation of the tangent line, then identified the limits of integration, set up the definite integral, and finally evaluated the integral to find the area.

The final answer is $\boxed{8/3}$, which corresponds to option (C).