#### To determine

**To show:**

The work done by the gas when the volume expands from volume V1 to volume V2 is

W=∫V1V2P dV

#### Answer

W=∫V1V2P dV

#### Explanation

**1) Concept:**

The work is calculated by using the formula W=∫abf(x)dx

**2) Given:**

The pressure is the function of volume P=P(V) and F=πr2P

**3) Calculation:**

Since the pressure P=P(V) is a function of volume, and the volume V=πr2x is a function of x, therefore, the pressure can be written as a function of x

Therefore, if V1=πr2x1 and V2=πr2x2 then the work done by the gas when the volume expands from volume V1 to volume V2 is

W=∫x1x2F(x) dx

Given F=πr2P

W=∫x1x2πr2P(V(x)) dx

Substitute (x)=πr2x, therefore,dV(x)=πr2dx and the limits changes as follows at x=x1, Vx=Vx1=V1 and at x=x2, Vx=Vx2=V2

W=∫V1V2P(V(x))dV(x)

W=∫V1V2P dV

Hence, it is shown that the work done by the gas when the volume expands from volume V1 to volume V2 is

W=∫V1V2P dV

**Conclusion:**

Thework done by the gas when volume expands from volume V1 to volume V2 is W=∫V1V2P dV