Daniel Alexander Murray

An Elementary Course in the Integral Calculus Daniel Alexander Murray

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. 1898 edition. Excerpt: ... Note 3. On putting a = b = c, the volume of a sphere of radius a is found to be $ ira3. Ex. 2. Find the volume of the ellipsoid given in Ex. 1: (a) by taking the infinitesimal slice at right angles to OY; (6) by taking it at right angles to OZ. Ex. 3. Determine the volume of a sphere of radius a by the method of this article. Ex. 4. Find the volume bounded by the hyperbolic paraboloid z = the xy-plane and the planes x = a, x = A, y = b, y = B.' Ex. 5. Find the volume of the wedge cut from the cylinder x2 + y2--a2 by the plane z = 0, and the part of the plane z = x tan a for which z is positive. Ex. 6. Find the entire volume bounded by the'surface Ex. 7. The center of a sphere of radius a is on the surface of a right ft cylinder the radius of whose base is--Find the volume of the part of the cylinder intercepted by the sphere. 63. Further application of successive integration to the measurement of volumes: polar coordinates. The illustration in this article is given, because the use of polar coordinates in dealing with solids is often advantageous. It will be necessary to employ these coordinates in solving some of the problems in Arts. 77, 79. Ex. 1. To find the volume of a sphere of radius a by means of polar coordinates. Let a point 0 on the surface of the sphere be the pole, the tan INTEGRAL CALC.--10 gent plane at 0 be the icy-plane, and the diameter through 0 be the 2-axis. Take any point P within the sphere, and let its coordinates be denoted by r, 6, j,--r being its distance OP from O, $ the angle that OP makes with the z-axis, and p the angle that the projection of OP on the xy-plane makes with the z-axis. Draw PM perpendicular to OZ. Produce OP an infinitesimal distance dr, and revolve OP through an infinitesimal angle dd in the...