Translated and adapted from a popular Russian educational series, this concise book requires only some background in high school algebra and elementary trigonometry. It explores the fundamental concept of the integral calculus: the limit of the sum of an infinitely increasing number of infinitely decreasing quantities. Mastery of this concept enables the solution of geometry and physics problems, and is an excellent introduction to the systematic study of higher mathematics. Starting with some algebraic formulas, the treatment proceeds to the determination of the pressure of a liquid on a vertical wall and the calculation of the work done in pumping liquid from a container. Subsequent chapters explore finding the volumes of a cone, pyramid, sphere, and other geometric forms and the measurement of the parabola, ellipse, and sinusoid. The text concludes with a selection of practice problems. For advanced high school students and college undergraduates.