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So I started by writing down the equation for the power transmitted to the gas in the container: [tex]P=-\oint p \vec{v} \cdot d\vec{A},[/tex] where the integral is taken over the entire surface of the container and v is the velocity of some point on the container. Assuming p is uniform, we get that

[tex]P=-p\oint \vec{v} \cdot d\vec{A} = -p\frac{d}{dt} \oint \vec{r} \cdot d\vec{A} = -p\frac{d}{dt} \int \nabla \cdot \vec{r} dV = -3p\frac{dV}{dt}.[/tex]

Integrating this equation with respect to time gives the wrong result by a factor of 3. What have I done wrong?