To find the area bounded by the parabola \( y = x^2 - 1 \), the tangent line at the point \( (2, 3) \), and the y-axis, we will follow these steps:
### Step 1: Find the slope of the tangent line
The equation of the parabola is given as \( y = x^2 - 1 \). To find the slope of the tangent line at the point \( (2, 3) \), we first differentiate the equation of the parabola.
\[
\frac{dy}{dx} = 2x
\]
Now, substituting \( x = 2 \):
\[
\text{slope} = 2(2) = 4
\]
### Step 2: Write the equation of the tangent line
Using the point-slope form of the line equation, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency \( (2, 3) \):
\[
y - 3 = 4(x - 2)
\]
Simplifying this:
\[
y - 3 = 4x - 8 \implies y = 4x - 5
\]
### Step 3: Find the intersection of the tangent line with the y-axis
To find the intersection with the y-axis, set \( x = 0 \):
\[
y = 4(0) - 5 = -5
\]
Thus, the point of intersection is \( (0, -5) \).
### Step 4: Determine the area between the parabola, tangent line, and y-axis
We need to find the area between the parabola \( y = x^2 - 1 \) and the tangent line \( y = 4x - 5 \) from \( x = 0 \) to \( x = 2 \).
The area \( A \) can be calculated using the integral:
\[
A = \int_{0}^{2} \left( (4x - 5) - (x^2 - 1) \right) dx
\]
Simplifying the integrand:
\[
A = \int_{0}^{2} (4x - 5 - x^2 + 1) dx = \int_{0}^{2} (-x^2 + 4x - 4) dx
\]
### Step 5: Evaluate the integral
Now we evaluate the integral:
\[
A = \int_{0}^{2} (-x^2 + 4x - 4) dx
\]
Calculating the integral:
\[
= \left[ -\frac{x^3}{3} + 2x^2 - 4x \right]_{0}^{2}
\]
Substituting the limits:
\[
= \left( -\frac{(2)^3}{3} + 2(2)^2 - 4(2) \right) - \left( -\frac{(0)^3}{3} + 2(0)^2 - 4(0) \right)
\]
Calculating the upper limit:
\[
= \left( -\frac{8}{3} + 8 - 8 \right) = -\frac{8}{3}
\]
The lower limit is zero, so:
\[
A = -\frac{8}{3} - 0 = -\frac{8}{3}
\]
Since area cannot be negative, we take the absolute value:
\[
A = \frac{8}{3}
\]
### Final Answer
Thus, the area bounded by the parabola, the tangent line, and the y-axis is:
\[
\boxed{\frac{8}{3}}
\]
