15. Find the volume of a prolate spheroid, whose transverse diameter is 7 feet 10 inches, and conjugate diameter is 3 feet 10 inches. 16. Find the volume of an oblong spheroid, whose transverse diameter is 8 feet 9 inches, and conjugate diameter is 4 feet 7 inches. PROBLEM 10. Guldinus's method of finding the volumes and Let G be the centre of gravity of the curved area H I, and draw G D perpendicular to A B. If the area HI revolve round A B as an axis, a solid of revolution will be generated. Guldinus has proved that the volume of 4 GIG this solid of revolution is equal to a prism, whose base is the revolving figure H I, and height the length described by the centre of gravity G. Let G' be the centre of gravity of the curved line H I. Then the surface of the solid of revolution is a rectangle, one side of which is the length of the generating curve H I, and the other the length described by the centre of gravity G'. This method is very useful in finding the volumes and surfaces of revolution, when the centre of gravity is readily found; and conversely, when the volumes and surfaces of revolution are known, the centres of gravity of the generating areas can readily be obtained. If H I be a circle, G is its centre, and the solid generated will be a cylindrical ring. G D I Length described by centre of gravity G = 16 X 3.1416. =50.2656. Volume of ring = 12.5664 X 50 2656 = 631 659 cubic feet. To find the surface of the ring, the centre of gravity of the area and the circumference of H I is evidently in the centre. = The length of the circumference HI 3·1416 x 412-5664 feet. Length described by the centre of gravity = 16 × 3·1416 = 50 2656 feet. Surface of volume 12 5664 X 50·2656 = 631 659 square feet. In this case the surface of the ring is equal to its volume. EXAMPLES. Section 1. 1. Find the volume and surface of a ring, whose thickness is 3 inches, and inner diameter 16 inches. 2. Find the volume and surface of a ring, whose thickness is 6 inches, and inner diameter 9 inches. Section 2. 3. Find the weight of a ring, of wrought iron, whose thickness is 1 inches, and inner diameter 6 inches. 4. Find the volume and surface of a ring, whose thickness is 8 inches, and inner diameter 18 inches. If h = 0, then the figure becomes a circular spindle, whose The following is an approximate formula instead of B, and in most cases it will not be far from the truth. 5. Find the number of gallons in a cask, whose bung and head diameters are 18 feet 10 inches, and length 24 inches. 6. Find the number of gallons in a cask, whose bung and head diameters are 20 and 14 inches, and length 38 inches. Section 4. 7. Find the volume of a circular spindle, whose height is 5 feet, and base 26 feet. 8. Find the number of gallons in a cask, whose bung and head diameters are 24 and 18 inches, and length 48 inches. Surface of D B PE=2r.CD-a (r—BC) BC -BC)} BP. In a right cone CBD E is the segment of a circle, whose diameter is A B; and CFDE is an ellipse, parabola, or hyperbola, according as FE B is less, equal to, or greater than GAB. |