An Elementary Course of Infinitesimal Calculus (Paperback)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 edition. Excerpt: ...as the sum or as the difference of two prisms or cylinders each having one end perpendicular to the length. We infer that in all cases the volume is equal to the area of the cross-section multiplied by the distance between the mass-centres of the two ends. Ex. The volume of the wedge-shaped solid cut off from a right circular cylinder by a plane through the centre of the base, making an angle a with the plane of the base, is 4 Jn-a2 x g-a tan a = o8 tan a; cf. Art. 114, Ex. 1. The theorems of Pappus may be generalized in various ways; but it may be sufficient here to state the following extension of the second theorem. If a plane area, constant or continuously variable, move about in any manner in space, but so that consecutive positions of the plane do not intersect within the area, the volume generated is equal to JSda (2), where S is the area, and da-is the projection of an element of the locus of the mass-centre of the area on the normal to the plane. If ds denote an element of this locus, and 0 the angle between ds and the normal to the plane, the formula may also be written fS cos dds (3). This theorem is the three-dimensional analogue of the proposition of Art. 101, relating to the area swept over by a moving line. It is a simple corollary from the theorem above proved. EXAMPLES. XXXVm. 1. A quadrant of a circle revolves about the tangent at one extremity; prove that the distance of the mass-centre of the curved surface generated, from the vertex, is-876a. 2. The mass-centre of either half of the surface of an anchor-ring cut off by the equatorial plane is at a distance 2bjir from this plane, where b is the radius of the generating circle. 3. Two equal circular holes of angular radius a are made in a uniform thin spherical shell, ..