Key idea: In a cyclic process on a $P$–$V$ diagram, the net work done by the gas over one complete cycle equals the area enclosed by the cycle.

From the figure, the cycle is a circle/ellipse-like closed curve on the $P$–$V$ plane with semi-axes:

• along volume axis: $2 \times 10^{-3}\,\text{m}^3$
• along pressure axis: $5 \times 10^{4}\,\text{Pa}$

Hence enclosed area is $W = \pi ab$ where $a = 2 \times 10^{-3}\,\text{m}^3, \qquad b = 5 \times 10^4\,\text{Pa}$

Substitute values: $W = 3.14 \times (2 \times 10^{-3}) \times (5 \times 10^4)$

Calculate: $(2 \times 10^{-3})(5 \times 10^4) = 10 \times 10^1 = 100$

So, $W = 3.14 \times 100 = 314\,\text{J}$

The question asks in the form: $\underline{\hspace{1cm}} \times 10^{-1}\,\text{J}$

Write $314\,\text{J}$ as: $314\,\text{J} = 3140 \times 10^{-1}\,\text{J}$

Therefore, the required integer to fill in the blank is $\boxed{3140}$

Comparison with stored answer:

• Stored answer = $314$
• Derived answer = $3140$

So the stored answer appears to have missed the conversion into the requested "$\times 10^{-1}\,\text{J}$" format.