An elementary treatise on the integral calculus, containing applications to plane curves and surfaces |
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Common terms and phrases
a₁ a²x² Accordingly algebraic function angle assume axis becomes infinite centre circle closed curve co-ordinates cone Consequently constant corresponding cos² cx² cycloid cylinder definite integral denote differentiate e-ax easily seen elementary ellipse epitrochoid equal equation Eulerian Integral evanescent EXAMPLES expression find the area find the volume finite formula function Gamma-Functions Hence hyperbola immediately integrable length limaçon limits method mx dx negative ordinates parabola paraboloid partial fractions pedal perpendicular plane polygon positive integer prove radius rectangle reducible represented result revolution right line roots roulette sin² sphere substitute suppose surface tan-¹ tangents theorem tractrix transformed vertex whole area x2 dx xm-1 dx ᏧᎾ
Popular passages
Page 212 - ... construction of an ellipse, as traced by a pencil which strains a thread passing over two fixed points, by substituting for those points a given ellipse, with which the locus described is confocal. This he deduced from the more general theorem on Spherical Conies ; the latter being arrived at from its reciprocal theorem, viz., if two spherical conies have the same cyclic arcs, then any arc touching the inner curve will cut off from the outer a segment of constant area. It may be here observed...
Page 228 - The volume of a cone is equal to one-third of the product of the base by the altitude.
Page 239 - J yds is the area of the surface generated by the curve, .-. &c. In like manner the second proposition can be shown to hold. Again, Guldin's theorems are still true if we suppose the rotation to take place around a number of different axes in succession ; in which case the centre of gravity, instead...
Page 238 - ... is equal to the product of the area of the generating curve into the path described by the center of gravity of the revolving area.
Page 79 - Further substitutions. Some integrals involving fractional powers of the variable x may be simplified by substituting x = zn, where n is the least common multiple of the denominators of the exponents. For example Vxdx I may be simplified by taking x = z*, dx = 4z3 dz.
Page 237 - P'Q' in a complete revolution is represented by 43Consequently the surface generated by the entire curve is 2irbS, where S denotes the whole length of the curve. A similar theorem holds for the volume of the solid generated : viz., the volume generated is equal to the product of the area of the revolving curve into the circumference of the same circle as before. For the volume of this solid is plainly represented by or by ir\(y- У') (У + У') dx = 2-nb \(y - y") dx.
Page 224 - CD shall be equal to a right line. 13. If a circle be described touching two tangents to an ellipse and also touching the ellipse, prove that the point of contact with the ellipse divides the elliptic arc between the points of contact of the tangents into two parts, whose •difference is equal to the difference of the lengths of the tangents (Chasles, Comptes Rendus, 1843). 14. Prove that the entire length of any closed curve is represented by...
Page 223 - I , and show that the area between the curve, the axis of x, and the ordinates at two points on the curve, is equal to a times the length of the arc terminated by those points.
Page 252 - EXAMPLES. 1. Find the area of the portion of the surface of a sphere which is intercepted by a right cylinder, one of whose edges passes through the centre of the sphere, and the radius of whose base is half that of the sphere.
Page 234 - I bounded by lines drawn parallel to the axis of x. The area of the surface generated by the revolution of a hyperbola round either axis admits of a similar investigation. EXAMPLES. i. Find the volume of the surface generated by the revolution of a cycloid round its base. Here, referring the cycloid to DA and NLB DB as co-ordinate axes, we have (see Diff.