A Treatise on the Integral Calculus and Its Applications with Numerous ExamplesMacmillan and Company, limited, 1878 - 408 páginas |
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Términos y frases comunes
1+p² angle Calculus of Variations closed curve co-ordinates constant cos² cycloid definite integral denote determine Differential Calculus differential coefficient double integral dx dx dx dy dz dx² dy dx equal equation example expression Find the area find the volume finite fixed point formula function given Hence infinite Integral Calculus integrate with respect latus rectum length limits maxima and minima obtain ordinates P₁ parabola partial fractions perpendicular plane preceding Article quantities radius vector result revolve round round the axis rx dx shew shewn sin² solid solid of revolution straight line suppose surface tangent theorem tion vanish variables vertex y₁ zero πα аф
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Página 175 - ... 5. Find the volume of the solid generated by the revolution of the closed part of the curve a?
Página 335 - From a point in the circumference of a circular field a projectile is thrown at random with a given velocity which is such that the diameter of the field is equal to the greatest range of the projectile : find the chance of its falling within the field.
Página 118 - ... r sin в cos ф, у = r sin в sin ф, z = r cos в...
Página 201 - О by a this perpendicular is ф, and the corresponding point, P, is found by the method described on page 255. Rectification of the Ellipse. 200. We have seen in Art. 177 that the length of an arc of an Ellipse measured from the end of the minor axis is s = If we let a;=asin<£, [1] becomes e = a (**Vi — e...
Página 352 - Prove that t • — =0, where s ds is the length of the arc of the curve measured from a fixed point on it. What is the geometrical meaning of — ? as 16. The expression for (ds)2 in parabolic coordinates is (ds)2 = (w2 + t>2)[(d«)2 + (dv)' What is the form assumed by Laplace's equation in these coordinates?
Página 334 - M the arithmetical mean of all the transverse sections of the solid generated by the revolution about the axis of x of the portion of the latter curve between the limits x = aв1 and x = aвt: shew that supposing ö2 greater than в1.
Página 367 - ... is to the area of the rectangle on the same base and of the same altitude as m : m + 1. This is equi/" valent to finding the value of I xmdx.
Página 391 - RESEARCHES IN THE CALCULUS OF VARIATIONS, principally on the Theory of Discontinuous Solutions: an Essay to which the Adams' Prize was awarded in the University of Cambridge in 1871.
Página 157 - Simpson's one-third rule," which is as follows : " Divide the base line into an even number of equal parts and erect ordinates at the points of division ; then add together the first and last ordinates, twice the sum of all the other odd ordinates, and four times the sum of all the even ordinates; multiply the sum by one-third of the common distance between ordinates.