Page:Calculus Made Easy.pdf/236

 to the work done in suddenly compressing the gas) from volume $$v_2$$ to volume $$v_1$$.

Here we have

An Exercise.

Prove the ordinary mensuration formula, that the area $$A$$ of a circle whose radius is $$R$$, is equal to $$\pi R^2$$.

Consider an elementary zone or annulus of the surface (Fig 59), of breadth $$dr$$, situated at a distance



$$r$$ from the centre. We may consider the entire surface as consisting of such narrow zones, and the whole area $$A$$ will simply be the integral of all