1680. The equation to the cycloid (origin at the vertex) being

show that the length of a cycloidal arc measured from the origin, is twice that of the chord of the corresponding arc of its generating circle.

1681. Find the area of the portion of a parabola cut off by the latus-rectum; and the volume of a conic frustum generated by the revolution of a rectangular trapezoid round its perpendicular side.

1682. Find the area of the curve y3 = a2. (x2 — b3) between the abscissas a and b.

1683. Find the area of the curve xy3 = 2a-x between the limits x = 0 and x = a.

1685.

limits x =

Find the area of the curve y3 = x3. (x − a)3 between the O and x = a.

1686. Find the area of the "Witch of Agnesi" bounded by the curve whose equation is xy=4r2 (2r-x), and by a straight line perpendicular to its axis and passing through the centre of the generating circle.

1687. Find the area of the curve y=x-x3 intercepted between the axes.

1688. Find the area of the curve

y. (4a2 + x3)=2(x + a) a3

between the limits x = 0 and x = 2a.

1689. Find the area of the curve expressed by the equation

1690. Find the volume of the solid generated by the revolution of the curve

round the axis of y.

xy = (a + x) (b3 — x3) $

1691. An ellipse revolves round a tangent at the extremity of the major axis; find the volume of the ring generated by the area of the semi-ellipse furthest from the tangent.

1692. Find the volume of the solid formed by the revolution of the curve xy3 = (a− x)3 round the axis of y, between the limits х = 0 and x = α,

=

1693. Let DC and CA be the semi-axes, minor and major, of an ellipse, and from any point E in the arc DA, draw EF parallel to AC and meeting DC in F; and let FC=h: it is required to find the area DFE, and the volume of a conoidal bullet generated by that area about FE.

1694. Find the volume of the solid generated by the revolution of the curve

y3 (a2 + x2) + a3x = a*

1695. Find the volume and curved surface of a paraboloid between the limits xa and x = b, the equation to the generating curve being y2 = 4mx.

1696. Find the centre of gravity of a quadrant of a circle.

1697. Find the centre of gravity of a circular sector of which the arc is 2a; and thence deduce that of a semicircular area.

1698. Find the centre of gravity of a segment of a circle in terms of the radius r of the circle, the semi-arc a of the segment, and the radius p of the base of the segment: and show what this becomes in the case of the semicircle.

1699. Find the centre of gravity of a semi-parabola of which the abscissa is a and the ordinate b.

1700. Find the centre of gravity of a circular arc a, in terms of the arc, its chord, and the radius of the circle.

1701. Find the centre of gravity of the solid generated by the revolution of the figure formed by two straight lines VA, AB at right angles to one another, and the parabolic arc VB about VA; when V is the vertex of the parabola of which the axis is parallel to AB.

1702. Find the centre of gravity of a material line, the density of which varies directly as the distance from one of its ends.

1703. Let the density of the sections of a right cone parallel to its base vary inversely as their distances from the vertex; find the centre of gravity.

1704. Find the centre of gravity of a cone, the density of every point of which varies inversely as the nth power of its distance from the plane of the base.

1705. Find the centre of gravity of the solid formed by the revolution of the area of the curve

about the axis of x; between the limits = 0, and x = a.

1706. At a point D, in an ellipse, the ordinate DH is equal

to the abscissa HC, C being the centre.

Find the centre of

gravity of the segment cut off by the double ordinate DIE.

1707. Let the density of a triangle vary as the nth power of the distance of any point in it from a straight line drawn through the vertex parallel to its base; find its centre of gravity.

1708. Let the density of a quadrant of a circle of uniform thickness vary as the nth power of the distance of any point in it from the centre of the circle; find its centre of gravity.

MOMENTS OF INERTIA.

Find the moment of inertia of

1709. A uniform rod about an axis through its centre of gravity and perpendicular to its length.

1710. A circle about an axis passing through its centre perpendicular to its plane.

arc.

1711. An equilateral triangle about one of its perpendiculars. 1712. A circular arc about the diameter which bisects the

1713. A circular arc about an axis passing through its vertex and perpendicular to its plane.

1714. The circumference of a circle about any tangent.

1715. A circular area about any diameter.

1716. A circular ring about an axis perpendicular to its plane passing through its centre.

1717. A cylinder about its axis.

1718. A sphere about any diameter.

1719. A right cone about its axis.

1720. A spherical shell about a diameter,

1721. A hollow cylinder about its axis.
1722. A spherical lamina about a diameter.

1723. A cylindrical lamina about its axis.

1724. A parallelogram about an axis perpendicular to its plane and passing through the intersection of its diagonals.

1725. A parabolic area about an axis perpendicular to its plane and passing through its vertex.

1726. A cube about its diagonal.

1727. A cube about the diagonal of one of its faces.

1728. A cube about one of its edges.

1729. A cone about its slant side.

1730. A spheroid about its axis of generation.

1731. An ellipsoid about one of its axes.

CENTRE OF OSCILLATION.

Find the time of a small oscillation of

1732. An equilateral triangle about an axis perpendicular to its plane, through one of its angles.

1733. A cube about one of its edges.

1734. A sphere about an axis touching its surface.

1735. A right cone about an axis touching the circumference of its base.

1736. Show that an arc of a circle will oscillate about an axis through its middle point perpendicular to its plane in the same time as if its mass were collected at the opposite extremity of the diameter of the complete circle.

1737. Show that a cylinder of 8 inches radius will oscillate about an axis on its surface parallel to its geometrical axis, in the same time as if its mass were collected at a point one foot distant from the axis.

1738. Show that a hemispherical surface will oscillate about a diameter of its base in the same time as a simple pendulum, the length of which is two-thirds of the diameter.