Ex. 1. The volume of the solid formed by the revolution of the cardioide whose equation is r = a (1+ cos 0), about the prime radius.

Ex. 2. The volume of the solid formed by the revolution of a sector of a circle, the angle of which a, about one of its Σπα bounding radii = 3

(1 cos a).

Ex. 3. The volume of the solid formed by the revolution of the circle whose equation is r = 2 a sin 0 about the prime radius = 22 a3.

Ex. 4. The volume of the solid formed by the revolution of the spiral of Archimedes, whose equation is ra0, with the

Ex. 5. If an ellipse revolves about its latus rectum, the volumes of the solids generated by the larger and smaller segments are respectively equal to

Hence, if an ellipse revolves about its latus rectum, the excess of volume generated by the larger segment over that generated by the smaller = 22ea2b.

Ex. 6. The volume formed by the revolution of the loop of the lemniscata, r2 = a2 cos 20, about its prime radius

Ex. 7. The vertex of a right cone, whose vertical angle is 2a, is on the surface of a sphere, and the axis of the cone passes through the centre; prove that the volume contained within the

255.] The surfaces which bound the volumes investigated in the preceding articles are all surfaces of revolution, and are therefore generated by circles whose planes are parallel, and whose radii vary according to a law assigned by the equation of the bounding curve; and the preceding method of cubature consists in the summation by the Integral Calculus of the circular slices into which the solid admits of being resolved: a similar method therefore is applicable to solids whose bounding surfaces are generated by curves or lines moving according to other given laws: the following cases exemplify the process.

Ex. 1. To find the volume of the elliptical cone, of a given altitude, which is generated by a varying ellipse moving parallel to itself and perpendicular to the axis of a, along which its centre moves, and the semi-axes of which are the ordinates to two straight lines intersecting at the origin and lying respectively in the planes of (z, x) and (x, y).

Let the equations to the lines be az = cx, ay = bx; then, as the area of an ellipse whose semi-axes are a and ẞ is aß, the volume of the elliptic elemental slice of the cone contained between two planes perpendicular to the axis of x, and at a distance пос a2

dx apart, is

x2 dx;

Ex. 2. Let the generating ellipse move as in the preceding example, and let its semi-axes be the ordinates of parabolas respectively in the planes of (z, x) and (x, y) whose equations are z2=4ax, y2 = 4 br; so that the volume of the elemental slice, whose thickness is dx, = 4π (ab) x dx;

therefore the volume of the solid = 4π(ab)

= 2π (ab)1 x2.

The bounding surface is the elliptic paraboloid.

x dx

Ex. 3. The volume of the ellipsoid may by a similar process

Ex. 4. Two equal quadrants of circles being described from the origin of coordinates as centre, and in the planes of (z, x) and (z, y), a variable square moving parallel to the plane of (x, y), and having the ordinates of the quadrants for sides, generates a surface called the Groin; it is required to determine the volume included between it and the coordinate planes.

The equations to the director-circles are, see fig. 38,

.. the volume of the square-element of the solid = (a2 — z2) dz ;

... the volume of the solid = [® (a2 — z2) dz

0

2a3 = 3

As the mode of generation may be continued into the other seven

octants, the volume of the whole groin =

16

3

α.

Ex. 5. A surface is generated by a rectangle moving parallel to the plane of (x, y), one of its sides being the ordinate of a given straight line passing through the origin which is in the plane of (y, z); and the other being an ordinate of a semicircle which is in the plane of (x, z), and passes through the origin, and whose diameter is coincident with the axis of z: it is required to find the volume of the solid.

Let the equations to the director-lines be

Ex. 6. On the double ordinates of an ellipse, and in planes perpendicular to that of the ellipse, isosceles triangles, of vertical angle 2a, are described; prove that the volume of the surface 4ab2 thus generated =

3 tan a

256.] If the generating plane area has not the axis of revolution for one of its containing sides, but is bounded by two curves whose equations are y = F(x) and y = f(x), then the elemental annular slice is equal to π {(F(x))2 — (ƒ (x))2} dx ;

and the volume required = dx #f{(x(x))2—(f(x))2} dæ ; (10)

the limits of integration being given by the geometry of the problem.

Of the formula (10) the following is an useful result. Suppose, as in fig. 39, that the generating plane, which for convenience sake we will call A, is such as to be divisible into two parts perfectly equal and symmetrical by a straight line parallel to ox, the axis of revolution, in the same manner as the closed figure EPCP' is divided by EC which is parallel to or: then, if the equation to EPC is y = f(x), when EBC is the axis of x, and if AB = a, the equations to EPC and EP'c with respect to or as the axis of x are severally ya+f(x), and y = a-f(x); and therefore by (10), the volume of the solid generated by the revolution of EPCP' about

that is, the volume of the solid is equal to the product of the revolving area and the circumference of the circle whose radius is the distance between the axis of revolution and that of symmetry.

Ex. 1. The volume generated by a circle of radius a about an axis in its own plane at a distance b from its centre = 2π2 a2 b. Ex. 2. The volume generated by an ellipse revolving about a tangent at the extremity of the major axis is 2m2 a2 b.

Ex. 3. The volume generated by an isosceles triangle, whose altitude a and base 26, about a line in its plane parallel to and at a distance c from the bisector of the vertical angle = 2πabс.

=

SECTION 2.-Cubature of Solids bounded by any Curved

Surface.

257.] If the bounding surface of the solid is not one of the particular forms discussed in the preceding section, but is determined by a general equation involving the coordinates of a point in space, the problem of cubature involves a triple integral, and is solved by the following process.

First, let position be determined by means of a system of rect

angular coordinates fixed in space; see fig. 40; and let E be a point within the space whose volume is to be found, and let E be (x, y, z); let F be another point within the space, and infinitesimally near to E, and let F be (x + dx, y+dy, z+dz); then the volume of the infinitesimal rectangular parallelepipedon of which E and F are two opposite angles is da dy dz: and the aggregate of all such, for the limits of integration assigned by the problem, is the required volume. If therefore v represents the required volume,

dz

= SS faz dy dx ;

V =

(12)

and the volume depends on the integration of this triple integral for the assigned limits.

Let us consider the effects of these successive integrations, and the relations between the limits of integration and the geometrical data; and, to fix our thoughts, let us suppose the volume to be such as that delineated in fig. 36, and contained within three coordinate planes.

Now dz dy dx is the volume of the infinitesimal parallelepipedon, one angle of which is at E, whose coordinates are x, y, z; the effect of the z-integration therefore, x and y not changing value, is the determination of the volume of a prismatic column, whose base is dy da, and whose altitude is given by the equations to the bounding surfaces: thus, if the volume is such as that delineated in fig. 36, and the equation to the surface is z = f(x, y), the limits of the z-integration are f(x, y) and 0; and the volume of the elemental column, whose height is NP, is f(x, y) dy dx: or if the volume to be determined is that contained between two surfaces whose equations are z = ƒ1 (x, y), z = ƒ2 (x, y) respectively, then the volume of the elemental column is

{f1 (x, y) —ƒ2 (x, y)} dy dx ;

(13)

and similarly whatever the bounding surfaces may be. Suppose that we next integrate with respect to y; the integral expressing the volume is now a double integral, and of the form

x

(14)

f(x, y) being a function of a and of y, which is introduced by the limits of z. Taking then fig. 36 to express the normal case, since f(x, y) dy dx is the volume of the elemental column, the sum of all such determined by the y-integration, when a is constant, is

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