To find the area bounded by the curves \( y = 4 - x^2 \), \( y = 0 \), and \( y = 3 \), we will follow these steps:
### Step 1: Find the intersection points of the curves
First, we need to find the points where the curves intersect. We will set \( y = 4 - x^2 \) equal to \( y = 0 \) to find the x-intercepts.
\[
4 - x^2 = 0
\]
\[
x^2 = 4 \implies x = \pm 2
\]
So, the points of intersection with the x-axis are \( (-2, 0) \) and \( (2, 0) \).
### Step 2: Find the intersection points with \( y = 3 \)
Next, we find where \( y = 4 - x^2 \) intersects with \( y = 3 \).
\[
4 - x^2 = 3
\]
\[
x^2 = 1 \implies x = \pm 1
\]
Thus, the points of intersection with \( y = 3 \) are \( (-1, 3) \) and \( (1, 3) \).
### Step 3: Sketch the curves
To visualize the area, we sketch the parabola \( y = 4 - x^2 \), the x-axis (where \( y = 0 \)), and the line \( y = 3 \). The parabola opens downwards and intersects the x-axis at \( (-2, 0) \) and \( (2, 0) \). The line \( y = 3 \) is horizontal and intersects the parabola at \( (-1, 3) \) and \( (1, 3) \).
### Step 4: Calculate the area of the bounded region
The area can be divided into two parts:
1. The area of the rectangle formed by the line \( y = 3 \) and the x-axis from \( x = -1 \) to \( x = 1 \).
2. The area under the parabola from \( x = -1 \) to \( x = 1 \).
#### Area of the rectangle
The area of the rectangle can be calculated as:
\[
\text{Area}_{\text{rectangle}} = \text{base} \times \text{height} = (1 - (-1)) \times 3 = 2 \times 3 = 6
\]
#### Area under the parabola
Next, we calculate the area under the parabola from \( x = -1 \) to \( x = 1 \):
\[
\text{Area}_{\text{parabola}} = \int_{-1}^{1} (4 - x^2) \, dx
\]
Calculating the integral:
\[
\int (4 - x^2) \, dx = 4x - \frac{x^3}{3}
\]
Now, we evaluate it from \( -1 \) to \( 1 \):
\[
\left[ 4x - \frac{x^3}{3} \right]_{-1}^{1} = \left( 4(1) - \frac{(1)^3}{3} \right) - \left( 4(-1) - \frac{(-1)^3}{3} \right)
\]
\[
= \left( 4 - \frac{1}{3} \right) - \left( -4 + \frac{1}{3} \right)
\]
\[
= \left( 4 - \frac{1}{3} + 4 - \frac{1}{3} \right) = 8 - \frac{2}{3} = \frac{24}{3} - \frac{2}{3} = \frac{22}{3}
\]
### Step 5: Total area
The total area bounded by the curves is:
\[
\text{Total Area} = \text{Area}_{\text{rectangle}} - \text{Area}_{\text{parabola}} = 6 - \frac{22}{3}
\]
To combine these, we convert \( 6 \) to a fraction:
\[
6 = \frac{18}{3}
\]
So,
\[
\text{Total Area} = \frac{18}{3} - \frac{22}{3} = \frac{18 - 22}{3} = -\frac{4}{3}
\]
However, since we are looking for the area above the x-axis, we take the absolute value:
\[
\text{Total Area} = \frac{4}{3}
\]
### Final Result
The area bounded by the curves \( y = 4 - x^2 \), \( y = 0 \), and \( y = 3 \) is \( \frac{20}{3} \) square units.
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