Find the area between the curve `y=x(x-3)` and the ordinates x = 3 and x = 5.

Answer

To find the area between the curve \( y = x(x - 3) \) and the ordinates \( x = 3 \) and \( x = 5 \), we will follow these steps:

### Step 1: Identify the function and the limits
The function given is:
\[ y = x(x - 3) = x^2 - 3x \]
We need to find the area between this curve and the x-axis from \( x = 3 \) to \( x = 5 \).

### Step 2: Set up the integral
The area \( A \) between the curve and the x-axis from \( x = 3 \) to \( x = 5 \) can be calculated using the definite integral:
\[ A = \int_{3}^{5} (x^2 - 3x) \, dx \]

### Step 3: Compute the integral
Now we will compute the integral:
\[ A = \int_{3}^{5} (x^2 - 3x) \, dx \]

First, we find the antiderivative of \( x^2 - 3x \):
\[ \int (x^2 - 3x) \, dx = \frac{x^3}{3} - \frac{3x^2}{2} \]

Now we evaluate this from \( 3 \) to \( 5 \):
\[ A = \left[ \frac{x^3}{3} - \frac{3x^2}{2} \right]_{3}^{5} \]

### Step 4: Calculate the definite integral
Now we substitute the limits into the antiderivative:
\[ A = \left( \frac{5^3}{3} - \frac{3 \cdot 5^2}{2} \right) - \left( \frac{3^3}{3} - \frac{3 \cdot 3^2}{2} \right) \]

Calculating the first part:
\[ \frac{5^3}{3} = \frac{125}{3}, \quad \frac{3 \cdot 5^2}{2} = \frac{75}{2} \]

Calculating the second part:
\[ \frac{3^3}{3} = 9, \quad \frac{3 \cdot 3^2}{2} = \frac{27}{2} \]

Now substituting these values back:
\[ A = \left( \frac{125}{3} - \frac{75}{2} \right) - \left( 9 - \frac{27}{2} \right) \]

### Step 5: Simplify the expression
To simplify, we need a common denominator. The least common multiple of 3 and 2 is 6:
\[ \frac{125}{3} = \frac{250}{6}, \quad \frac{75}{2} = \frac{225}{6} \]
Thus,
\[ \frac{250}{6} - \frac{225}{6} = \frac{25}{6} \]

For the second part:
\[ 9 = \frac{54}{6}, \quad \frac{27}{2} = \frac{81}{6} \]
Thus,
\[ \frac{54}{6} - \frac{81}{6} = -\frac{27}{6} \]

Now, combining these results:
\[ A = \frac{25}{6} - \left(-\frac{27}{6}\right) = \frac{25}{6} + \frac{27}{6} = \frac{52}{6} = \frac{26}{3} \]

### Final Answer
The area between the curve \( y = x(x - 3) \) and the ordinates \( x = 3 \) and \( x = 5 \) is:
\[ \boxed{\frac{26}{3}} \]