Find the area of the region bounded by
`x^(2)=8y`, x-axis and the line x= 4

Answer

To find the area of the region bounded by the parabola \( x^2 = 8y \), the x-axis, and the line \( x = 4 \), we will follow these steps:

### Step 1: Rewrite the equation of the parabola
The equation of the parabola is given by \( x^2 = 8y \). We can rewrite this in terms of \( y \):
\[ y = \frac{x^2}{8} \]

**Hint:** Remember that the parabola opens upwards, and we want to express \( y \) in terms of \( x \) for integration.

### Step 2: Identify the limits of integration
The area we want to find is bounded by the x-axis (where \( y = 0 \)), the parabola, and the vertical line \( x = 4 \). The intersection of the parabola with the x-axis occurs when \( y = 0 \):
\[ 0 = \frac{x^2}{8} \implies x^2 = 0 \implies x = 0 \]
Thus, the limits of integration are from \( x = 0 \) to \( x = 4 \).

**Hint:** Check where the curve intersects the x-axis to determine the lower limit.

### Step 3: Set up the integral for the area
The area \( A \) under the curve from \( x = 0 \) to \( x = 4 \) can be expressed as:
\[ A = \int_{0}^{4} y \, dx = \int_{0}^{4} \frac{x^2}{8} \, dx \]

**Hint:** The area under the curve is found by integrating the function representing the curve.

### Step 4: Calculate the integral
Now we will compute the integral:
\[ A = \int_{0}^{4} \frac{x^2}{8} \, dx = \frac{1}{8} \int_{0}^{4} x^2 \, dx \]
The integral of \( x^2 \) is:
\[ \int x^2 \, dx = \frac{x^3}{3} \]
Thus, we have:
\[ A = \frac{1}{8} \left[ \frac{x^3}{3} \right]_{0}^{4} \]

**Hint:** Remember to evaluate the integral at the upper and lower limits.

### Step 5: Evaluate the definite integral
Now we substitute the limits into the integral:
\[ A = \frac{1}{8} \left( \frac{4^3}{3} - \frac{0^3}{3} \right) = \frac{1}{8} \left( \frac{64}{3} - 0 \right) = \frac{64}{24} = \frac{8}{3} \]

**Hint:** Simplify the fraction carefully to find the final area.

### Final Answer
The area of the region bounded by the parabola \( x^2 = 8y \), the x-axis, and the line \( x = 4 \) is:
\[ \boxed{\frac{8}{3}} \text{ square units} \]