Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Результати 6-10 із 45
Сторінка 416
... limits . Integrating with respect to r we have A = 1 fr2 do + C ; and if we suppose C = 0 , the integral A = fr2 do , in which there is substituted for r its value in terms of given by the equation to the curve , is the value of the ...
... limits . Integrating with respect to r we have A = 1 fr2 do + C ; and if we suppose C = 0 , the integral A = fr2 do , in which there is substituted for r its value in terms of given by the equation to the curve , is the value of the ...
Сторінка 419
... the loop . ( 22 ) The lemniscate whose equation is ( x2 + y2 ) 2 = a2x2 — b3y3 , - has two loops ; find its area . a2 - b2 t2 A = 1⁄2 [ dt x2 = 1 [ dt ( 1 + 12 ) ? a b and for each loop the limits of t 27-2 QUADRATURE OF AREAS . 419.
... the loop . ( 22 ) The lemniscate whose equation is ( x2 + y2 ) 2 = a2x2 — b3y3 , - has two loops ; find its area . a2 - b2 t2 A = 1⁄2 [ dt x2 = 1 [ dt ( 1 + 12 ) ? a b and for each loop the limits of t 27-2 QUADRATURE OF AREAS . 419.
Сторінка 420
... limits . ( 1 ) The equation to the common parabola being y3 = 4mx , the length of an arc measured from the vertex is = m m + 2 { x + ( x2 + mx ) 1 sdæ ( * + TM ) * = ( x2 + mx ) 3 + ( x m ) - log ( 2 ) m The equation to the semicubical ...
... limits . ( 1 ) The equation to the common parabola being y3 = 4mx , the length of an arc measured from the vertex is = m m + 2 { x + ( x2 + mx ) 1 sdæ ( * + TM ) * = ( x2 + mx ) 3 + ( x m ) - log ( 2 ) m The equation to the semicubical ...
Сторінка 424
... limits , which differ according to the nature of the surfaces which bound the surface laterally . The most simple case is when the solid is bounded laterally by four planes , two parallel to the plane of wx , and two to that of y . The ...
... limits , which differ according to the nature of the surfaces which bound the surface laterally . The most simple case is when the solid is bounded laterally by four planes , two parallel to the plane of wx , and two to that of y . The ...
Сторінка 425
... limits of the integral with respect to y . There remains now only to integrate a function of x , and to take it between the limits of the value of a derived from the equation to the surface by making = 0 and y = 0 . If we wish to find ...
... limits of the integral with respect to y . There remains now only to integrate a function of x , and to take it between the limits of the value of a derived from the equation to the surface by making = 0 and y = 0 . If we wish to find ...
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a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz dz dz eliminate ellipse equal Euler factor formula fraction function Geometry gives Hence hypocycloid infinite intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Multiply negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle University of Cambridge vanish whence x²)³