being the limits of yo being 0 and a cos 0, and those of O anda. Hence Therefore the whole solid cut out from the hemisphere is 2* a - ***; and the part of the hemisphere which is not comprised in the cylinder is soor - of the cube of the diameter of the sphere. (8) A paraboloid of revolution is pierced by a right circular cylinder, the axis of which passes through the focus and cuts the axis at right angles, its radius being one fourth of the latus rectum of the generating parabola ; find the volume of the solid common to the two surfaces. The equations to the surfaces are y2 + x = 4ax, 2 + y = 2ax. Hence V = ssdx dy (4a x – yo)? When a solid is generated by the motion of a plane area which moves parallel to itself, while its magnitude increases or decreases according to a given law, its volume is found by the formula V = cos a su dx; v being the area, the axis of x being the direction of motion, and making a constant angle a with the normal to the plane. (9) Let the solid be the groin which is generated by a square moving parallel to itself, its sides being the double (10) Find the content of the solid ABCDO (fig. 55); the base ABCD being a rectangle, the side OAB a rightangled triangle perpendicular to the plane of the rectangle, and the upper side OBCD being formed by drawing lines as PQ from OB to CD, always parallel to the plane OAD. If we draw PR parallel to 0 A, and join RQ, the triangle PQR having two sides parallel to the sides of ODA, is in a plane parallel to that of ODA. Hence the figure may be supposed to be generated by the motion of a triangle constantly parallel to AOD, and having its angular points in the lines AB, OB, CD. If AD = a, AB = b, AO = C, the abc volume of the solid ise. (11) The axes of two equal right circular cylinders intersect at an angle a; to find the volume of the solid common to both. Let ABCD (fig. 56) be the section of the solid made by the plane containing the axes, and let the radius of the cylinders = a, so that AB = a cosec a. If we cut the solid by a plane parallel to ABCD, we shall have a parallelogram as PQRS; and calling the area of this A, and its distance from the plane of the axes x, we shall have for the part of the solid above that plane V = ." dz A. Now A = 4 POQ; but making PQ = l, and calling p the perpendicular on PQ from the point in the plane PQRS where it meets a line through () perpendicular to the plane of the axes, we have l= p (tan} a + cot? a) = 2 p cosec a, and therefore POQ = p’ cosec a, and A = 4 p’ cosec a. But the section through O and the perpendicular p being a semicircle, we have på = a’ – . Hence V = 4 cosec a “ (a' – **) dx = f a cosec a, . 16 a and therefore the whole solid is 3 sina: (12) Find the volume of the solid DEQB (fig. 57) cut off from a right circular cylinder by a plane EQD passing through the centre of the base, and inclined at an angle a to the plane of the base. If we cut the solid by a plane perpendicular to the base of the cylinder, and parallel to the trace ED, the section is a parallelogram, and the solid may be considered as generated by the motion of this parallelogram parallel to itself. Let CB = a, CM = x, MN = Y, PN = x, then as x = x tan a, and y = (a’ – x*)!, V = 2 tan a sdx x (a’ – 20*)! = tan a {C – (a® – 20°)}}. When x = 0, W = 0, therefore C = a'; hence V = s tan a {a' – (a' – 2')}; and the whole solid when x = a is a' tan a. If the solid be one of revolution round the axis of , and if y = f (x) be the equation to the generating curve, the volume of the solid is given by the integral V = Syda. (13) A paraboloid formed by the revolution of a parabola round its axis, In this case y' = 4mx, and V = 4m 7 fæ dx = 2 m 7 M2 + C. V = 2 ml 9x = 1 yox. (14) The volume of an oblate spheroid formed by the revolution of an ellipse round its minor axis is - a being the major axis of the ellipse; and the volume of a prolate spheroid is * a: 47 ab (15) Find the volume of the solid formed by the revolution of the cissoid round its asymptote. The asymptote being taken as the axis of X, the equation to the cissoid is x+y = (2a - y), a being the radius of the generating circle. (17) The equation to the cycloid is (the base being the axis of w), Therefore the volume formed by its revolution round the base is daca dy T2ay V = a sdy '(2a 4 - 4)^" This being integrated from y = 0 to y = 2a and doubled gives 5 7° a: as the volume of the whole solid. When the axis of the cycloid is taken as the axis of X, the equation to the curve is dy (2a - x d x 1 x . but this is not a convenient form for finding the value of a sydx. It is better to substitute for y and x their expressions in terms of 0, i. e. y = a (0 + sin o), X = a (1 – cos 0); whence V = qar sin 0 (0 + sin 0)”. The value of this taken from 0 = 0 to 0 = , is ce (18) The equation to the tractory is (6o – y): dy + y = 0; and the volume of the solid generated by its revolution round the axis of X, and taken from X = 0 to x = c is ma’. (19) The equation to the Witch of Agnesi is xy = 2a (2ay – y'). |