Find the area of the region bounded by `y^(2)=4ax` between the lines x=a and x= 9a

Answer

To find the area of the region bounded by the parabola \( y^2 = 4ax \) between the lines \( x = a \) and \( x = 9a \), we can follow these steps:

### Step 1: Understand the equation of the parabola
The equation \( y^2 = 4ax \) represents a parabola that opens to the right. To express \( y \) in terms of \( x \), we take the square root:
\[ y = \sqrt{4ax} \quad \text{and} \quad y = -\sqrt{4ax} \]
This means the parabola has two branches, one above the x-axis and one below.

### Step 2: Set up the integral for the area
The area \( A \) between the curves from \( x = a \) to \( x = 9a \) can be calculated using the integral of the upper curve minus the lower curve. Since the upper curve is \( y = \sqrt{4ax} \) and the lower curve is \( y = -\sqrt{4ax} \), we can express the area as:
\[ A = \int_{a}^{9a} \left( \sqrt{4ax} - (-\sqrt{4ax}) \right) \, dx \]
This simplifies to:
\[ A = \int_{a}^{9a} 2\sqrt{4ax} \, dx \]

### Step 3: Simplify the integral
We can factor out the constant from the integral:
\[ A = 2 \int_{a}^{9a} \sqrt{4ax} \, dx = 2 \int_{a}^{9a} 2\sqrt{ax} \, dx = 4 \int_{a}^{9a} \sqrt{ax} \, dx \]

### Step 4: Change the variable for integration
Let \( k = \sqrt{a} \), then \( \sqrt{ax} = k\sqrt{x} \). The integral becomes:
\[ A = 4 \int_{a}^{9a} k\sqrt{x} \, dx \]
Now, we can integrate \( \sqrt{x} \):
\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} \]
Thus,
\[ A = 4k \cdot \frac{2}{3} \left[ x^{3/2} \right]_{a}^{9a} \]

### Step 5: Evaluate the definite integral
Now we will evaluate the limits:
\[ A = \frac{8k}{3} \left[ (9a)^{3/2} - a^{3/2} \right] \]
Calculating \( (9a)^{3/2} \):
\[ (9a)^{3/2} = 27a^{3/2} \]
Thus, we have:
\[ A = \frac{8k}{3} \left[ 27a^{3/2} - a^{3/2} \right] = \frac{8k}{3} \cdot 26a^{3/2} = \frac{208a^{3/2}}{3} \]

### Step 6: Substitute back for \( k \)
Since \( k = \sqrt{a} \), we substitute back:
\[ A = \frac{208a^{3/2}}{3} \]

### Final Result
The area of the region bounded by the parabola \( y^2 = 4ax \) between the lines \( x = a \) and \( x = 9a \) is:
\[ \boxed{\frac{208a^2}{3}} \]