The prssure in a monoatomic gas increases lineraly from `4 xx 10^(5) Nm^(-2) to 8 xx 10^(5)Nm^(-2)` when its volume increases form `0.2 m^(3) to 0.5 m^(3)`. Calculate the following:
(a) work done by the gas. (2) increase in the internal energy

Explanation

To solve the problem, we need to calculate the work done by the gas and the increase in internal energy when the pressure of a monoatomic gas increases linearly from \(4 \times 10^5 \, \text{N/m}^2\) to \(8 \times 10^5 \, \text{N/m}^2\) as its volume increases from \(0.2 \, \text{m}^3\) to \(0.5 \, \text{m}^3\).

### Step 1: Calculate the Work Done by the Gas

The work done by the gas during an expansion can be found by calculating the area under the pressure-volume (P-V) graph. Since the pressure increases linearly, the graph will form a trapezium.

1. **Identify the points on the graph:**
- Point A: \( (V_1, P_1) = (0.2, 4 \times 10^5) \)
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