Using integration, find the area bounded between the parabola `x ^(2) = 4y ` and the line `y=4.`

Answer

To find the area bounded between the parabola \(x^2 = 4y\) and the line \(y = 4\), we can follow these steps:

### Step 1: Identify the curves and points of intersection
The given equations are:
- Parabola: \(x^2 = 4y\)
- Line: \(y = 4\)

To find the points of intersection, we substitute \(y = 4\) into the parabola's equation:
\[ x^2 = 4(4) = 16 \implies x = \pm 4 \]
Thus, the points of intersection are \((4, 4)\) and \((-4, 4)\).

### Step 2: Set up the integral for the area
The area \(A\) between the curves from \(x = -4\) to \(x = 4\) can be calculated using the formula:
\[ A = \int_{-4}^{4} (y_{\text{line}} - y_{\text{parabola}}) \, dx \]
Here, \(y_{\text{line}} = 4\) and \(y_{\text{parabola}} = \frac{x^2}{4}\) (from rearranging \(x^2 = 4y\)).

Thus, the area becomes:
\[ A = \int_{-4}^{4} \left(4 - \frac{x^2}{4}\right) \, dx \]

### Step 3: Simplify the integral
We can simplify the integrand:
\[ A = \int_{-4}^{4} \left(4 - \frac{x^2}{4}\right) \, dx = \int_{-4}^{4} \left(\frac{16}{4} - \frac{x^2}{4}\right) \, dx = \int_{-4}^{4} \frac{16 - x^2}{4} \, dx \]
This can be factored out:
\[ A = \frac{1}{4} \int_{-4}^{4} (16 - x^2) \, dx \]

### Step 4: Calculate the integral
Now we compute the integral:
\[ A = \frac{1}{4} \left[ \int_{-4}^{4} 16 \, dx - \int_{-4}^{4} x^2 \, dx \right] \]
Calculating the first integral:
\[ \int_{-4}^{4} 16 \, dx = 16 \cdot (4 - (-4)) = 16 \cdot 8 = 128 \]
Calculating the second integral:
\[ \int_{-4}^{4} x^2 \, dx = 2 \int_{0}^{4} x^2 \, dx = 2 \left[\frac{x^3}{3}\right]_{0}^{4} = 2 \left[\frac{64}{3} - 0\right] = \frac{128}{3} \]

### Step 5: Combine the results
Now substituting back into the area formula:
\[ A = \frac{1}{4} \left(128 - \frac{128}{3}\right) \]
Finding a common denominator:
\[ 128 = \frac{384}{3} \implies A = \frac{1}{4} \left(\frac{384}{3} - \frac{128}{3}\right) = \frac{1}{4} \left(\frac{256}{3}\right) = \frac{256}{12} = \frac{64}{3} \]

### Final Answer
Thus, the area bounded between the parabola \(x^2 = 4y\) and the line \(y = 4\) is:
\[ \boxed{\frac{64}{3}} \]