A Treatise on the Integral Calculus and Its Applications with Numerous ExamplesMacmillan, 1862 - 342 Seiten |
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1+p² 1+x² Calculus of Variations co-ordinates constant cos² curve cycloid definite integral denote determine differential coefficient double integral dv dv dx dx dx dy dz dx² dy dv dy dx dy equal equation example expression Find the area find the volume fixed point formula function given Hence Integral Calculus integrate with respect latus rectum left-hand member length limits maxima and minima maximum or minimum obtain ordinate P₁ parabola partial fractions plane positive preceding article quantity radius vector result revolution revolve round round the axis rx dx shew sin² solid solid of revolution suppose surface tangent tion vanish variables vertex x₁ Y₁ zero πα аф
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Seite 167 - EXAMPLE. A right cone has its vertex on the surface of a sphere, and its axis coincident with the diameter of the sphere passing through that point ; find the volume common to the cone and the sphere. Volume of any Solid. Triple Integration.
Seite 305 - Let s denote the length of the arc of the curve measured from a fixed point; then, by integrating the last equation, we have This shews that the required curve is a cycloid; see Art.
Seite 53 - I sin2 в d6 . : о ,o follows immediately from the definition of integration, and the fact that the sine of an angle is equal to the sine of the supplemental angle.
Seite 309 - In geometrical problems \j-\ is the tangent of the inclination to the axis of x of the...
Seite 342 - Pot 8vo. [In the Press. Algebra. For the use of Colleges and Schools. By I. TODHUNTER, MA. Fellow of St. John's College, Cambridge. Second Edition. Crown 8vo.
Seite 180 - This gives the result of Art. 216. 219. Example. Find a curve such that the area between the curve, the axis of x, and any ordinate, shall bear a constant ratio to the rectangle contained by that ordinate and the corresponding abscissa. Suppose ф (x) the ordinate of the curve to the abscissa x ; then I ф (x) dx expresses the area between the curve, the axis of x, and the ordinate ф (с) : hence by supposition we must have сф (с) I.
Seite 149 - A sphere is pierced perpendicularly to the plane of one of its great circles by two right cylinders, of which the diameters are equal to the radius of the sphere and the axes pass through the middle points of two radii that compose a diameter of this great circle. Find the surface of that portion of the sphere not included within the cylinders.