Find the area bounded under the curve `y=3x^2+6x+7` X-axis with the oridinateks at x=5 and x=10.

Explanation

To find the area bounded under the curve \( y = 3x^2 + 6x + 7 \) and the x-axis between the ordinates at \( x = 5 \) and \( x = 10 \), we will use definite integration.

### Step-by-step Solution:

1. **Set up the integral**:
The area \( A \) under the curve from \( x = 5 \) to \( x = 10 \) can be expressed as:
\[ A = \int_{5}^{10} (3x^2 + 6x + 7) \, dx \]

2. **Integrate the function**:
We will now integrate the function \( 3x^2 + 6x + 7 \):
\[ \int (3x^2 + 6x + 7) \, dx = 3 \cdot \frac{x^3}{3} + 6 \cdot \frac{x^2}{2} + 7x + C \]
Simplifying this gives:
\[ = x^3 + 3x^2 + 7x + C \]

3. **Evaluate the definite integral**:
Now we will evaluate the integral from \( x = 5 \) to \( x = 10 \):
\[ A = \left[ x^3 + 3x^2 + 7x \right]_{5}^{10} \]
This means we will calculate:
\[ A = \left(10^3 + 3 \cdot 10^2 + 7 \cdot 10\right) - \left(5^3 + 3 \cdot 5^2 + 7 \cdot 5\right) \]

4. **Calculate the upper limit**:
For \( x = 10 \):
\[ 10^3 = 1000, \quad 3 \cdot 10^2 = 300, \quad 7 \cdot 10 = 70 \]
Thus,
\[ 10^3 + 3 \cdot 10^2 + 7 \cdot 10 = 1000 + 300 + 70 = 1370 \]

5. **Calculate the lower limit**:
For \( x = 5 \):
\[ 5^3 = 125, \quad 3 \cdot 5^2 = 75, \quad 7 \cdot 5 = 35 \]
Thus,
\[ 5^3 + 3 \cdot 5^2 + 7 \cdot 5 = 125 + 75 + 35 = 235 \]

6. **Subtract the lower limit from the upper limit**:
Now, we find the area:
\[ A = 1370 - 235 = 1135 \]

7. **Final result**:
The area bounded under the curve \( y = 3x^2 + 6x + 7 \) and the x-axis between \( x = 5 \) and \( x = 10 \) is:
\[ A = 1135 \text{ square units} \]