This introduction to the theory of the Lebesgue Integral is primarily intended for third-year Honours students. A consistently geometrical approach has been chosen in order to show the simple underlying ideas. First, the volume of an n-dimensional set is defined as it s Lebesgue Measure. The integral of a non-negative function of n real variables is then the volume of its ordinate set; the extension to general functions follows easily. The old notion of the Riemann Integral is similarly developed; its relation to the Lebesgue Integral and the striking advantages of the latter are clearly set out.