Find the area of the region bounded by the curve `y =x ^(2)` and the line `y =4.`

Answer

To find the area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \), we can follow these steps:

### Step 1: Identify the points of intersection
To find the area between the curves, we first need to determine where they intersect. We set the equations equal to each other:

\[ x^2 = 4 \]

Taking the square root of both sides, we find:

\[ x = \pm 2 \]

Thus, the points of intersection are \( (-2, 4) \) and \( (2, 4) \).

**Hint:** Set the two equations equal to find the intersection points.

### Step 2: Set up the integral for the area
The area \( A \) between the curves from \( x = -2 \) to \( x = 2 \) can be calculated using the integral:

\[ A = \int_{-2}^{2} (4 - x^2) \, dx \]

Here, \( 4 \) is the upper curve (the line) and \( x^2 \) is the lower curve (the parabola).

**Hint:** The area is found by integrating the difference between the upper and lower curves.

### Step 3: Calculate the integral
Now we calculate the integral:

\[ A = \int_{-2}^{2} (4 - x^2) \, dx \]

This can be split into two separate integrals:

\[ A = \int_{-2}^{2} 4 \, dx - \int_{-2}^{2} x^2 \, dx \]

Calculating the first integral:

\[ \int_{-2}^{2} 4 \, dx = 4[x]_{-2}^{2} = 4(2 - (-2)) = 4 \times 4 = 16 \]

Now calculating the second integral:

\[ \int_{-2}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{-2}^{2} = \frac{2^3}{3} - \frac{(-2)^3}{3} = \frac{8}{3} - \left(-\frac{8}{3}\right) = \frac{8}{3} + \frac{8}{3} = \frac{16}{3} \]

Now substituting back into the area formula:

\[ A = 16 - \frac{16}{3} \]

To combine these, we convert \( 16 \) into a fraction with a denominator of \( 3 \):

\[ 16 = \frac{48}{3} \]

Thus,

\[ A = \frac{48}{3} - \frac{16}{3} = \frac{32}{3} \]

### Step 4: Final area calculation
Since the area is symmetric about the y-axis, we can also express the area as:

\[ A = 2 \int_{0}^{2} (4 - x^2) \, dx \]

Calculating this gives the same result:

\[ A = \frac{32}{3} \]

### Conclusion
The area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \) is:

\[ \boxed{\frac{32}{3}} \text{ square units} \]

---