Area between the parabola `y^(2)=9x` and the line `y=x` is …............

Answer

To find the area between the parabola \( y^2 = 9x \) and the line \( y = x \), we will follow these steps:

### Step 1: Find the Points of Intersection
To find the area between the two curves, we first need to determine the points where they intersect. We can do this by substituting \( y = x \) into the parabola's equation.

1. Substitute \( y = x \) into \( y^2 = 9x \):
\[ x^2 = 9x \]
Rearranging gives:
\[ x^2 - 9x = 0 \]
Factoring out \( x \):
\[ x(x - 9) = 0 \]
This gives us the solutions \( x = 0 \) and \( x = 9 \).

2. Now, substitute these \( x \) values back into \( y = x \) to find the corresponding \( y \) values:
- For \( x = 0 \), \( y = 0 \) → Point \( A(0, 0) \)
- For \( x = 9 \), \( y = 9 \) → Point \( B(9, 9) \)

### Step 2: Set Up the Area Integral
Next, we will set up the integral to find the area between the curves from \( x = 0 \) to \( x = 9 \).

1. The area \( A \) between the curves can be expressed as:
\[ A = \int_{0}^{9} (y_{\text{top}} - y_{\text{bottom}}) \, dx \]
Here, \( y_{\text{top}} \) is the line \( y = x \) and \( y_{\text{bottom}} \) is the parabola \( y = 3\sqrt{x} \) (since \( y^2 = 9x \) implies \( y = \pm 3\sqrt{x} \), we take the positive root for the area calculation).

2. Thus, the area becomes:
\[ A = \int_{0}^{9} (x - 3\sqrt{x}) \, dx \]

### Step 3: Calculate the Integral
Now we will compute the integral.

1. Split the integral:
\[ A = \int_{0}^{9} x \, dx - \int_{0}^{9} 3\sqrt{x} \, dx \]

2. Calculate each integral separately:
- For \( \int_{0}^{9} x \, dx \):
\[ \int x \, dx = \frac{x^2}{2} \bigg|_{0}^{9} = \frac{9^2}{2} - 0 = \frac{81}{2} \]

- For \( \int_{0}^{9} 3\sqrt{x} \, dx \):
\[ \int 3\sqrt{x} \, dx = 3 \cdot \frac{2}{3} x^{3/2} \bigg|_{0}^{9} = 2x^{3/2} \bigg|_{0}^{9} = 2(9^{3/2}) - 0 = 2(27) = 54 \]

3. Combine the results:
\[ A = \frac{81}{2} - 54 = \frac{81}{2} - \frac{108}{2} = \frac{-27}{2} \]

### Step 4: Final Area Calculation
Since area cannot be negative, we take the absolute value:
\[ A = \frac{27}{2} \]

### Conclusion
The area between the parabola \( y^2 = 9x \) and the line \( y = x \) is:
\[ \boxed{\frac{27}{2}} \text{ square units.} \]