This is a companion volume to Professor Lodge's "Differential Calculus for Beginners." In that volume the student was prepared to practice retracing his steps, and thus, without the use of the integral notation, to perform the operation of integration or anti-differentiation in simple cases. Hence the author is in a position to commence this volume by exhibiting an integral as the limit of a sum; and that no time is wasted in getting to business is evidenced by the fact that the centre of gravity of a parabolic area is worked out at p. 9. The standard methods of integration are clearly explained and illustrated in the first five chapters. The most novel feature of the book is perhaps the seventh chapter dealing with approximate methods of integration. Here, after the well-known rules of Simpson and Weddle, approximate formulae, recently devised by Mr. R. W. K. Edwards and Professor Lodge himself, are given, for dealing with the case in which the curvilinear boundary of a required area cuts the axis at right angles; a case for which, as is well known, rules of the Simpson type are not well fitted.