the linear charge density of a thin metallic rod varies with the distance x from the end as `lambda=lambda_(0)x^(2)(0 le x lel)` The total charge on the rod is

Explanation

To find the total charge on the thin metallic rod with a linear charge density that varies with distance, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Linear Charge Density**:
The linear charge density is given by the equation:
\[ \lambda(x) = \lambda_0 x^2 \]
where \( \lambda_0 \) is a constant and \( x \) varies from 0 to \( L \).

2. **Define a Small Element of Charge**:
Consider a small element of the rod at a distance \( x \) with a small length \( dx \). The charge \( dQ \) on this small element can be expressed as:
\[ dQ = \lambda(x) \cdot dx = \lambda_0 x^2 \cdot dx \]

3. **Integrate to Find Total Charge**:
To find the total charge \( Q \) on the rod, we need to integrate \( dQ \) from \( x = 0 \) to \( x = L \):
\[ Q = \int_{0}^{L} dQ = \int_{0}^{L} \lambda_0 x^2 \, dx \]

4. **Perform the Integration**:
The integral can be computed as follows:
\[ Q = \lambda_0 \int_{0}^{L} x^2 \, dx \]
The integral of \( x^2 \) is:
\[ \int x^2 \, dx = \frac{x^3}{3} \]
Therefore, we have:
\[ Q = \lambda_0 \left[ \frac{x^3}{3} \right]_{0}^{L} = \lambda_0 \left( \frac{L^3}{3} - 0 \right) = \frac{\lambda_0 L^3}{3} \]

5. **Final Result**:
Thus, the total charge \( Q \) on the rod is:
\[ Q = \frac{\lambda_0 L^3}{3} \]