## A treatise on the integral calculus and its applications1857 |

### From inside the book

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**find the area**included between this curve , the axis of x , and the ordinates corresponding to the abscissę a and b . Let OA = a , OB = b ; divide the space AB into n equal intervals and draw ordinates at the points of division ... Page 104

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**find**the whole**area**of the ellipse . b2 The equation to the ellipse may be written y2 = — ( a2 — x2 ) . Hence the**area**of one quadrant of the ellipse b ' b ra = [ ] " { √ ( a2 — xo ) dx = - a √ √ √ ( a2 = x2 ) dx 0 - = - a 0 - hence ... Page 109

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**find**it easier to apply the processes of double integration to those cases where it is absolutely necessary , of which examples will occur here- after . 139. If the**area**which is to be evaluated is bounded by the curves x = ( y ) , and ... Page 111

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**area**into two parts by the line AF . Let - Y1 = √ ( 4a2 — 4ax ) , Y2 = √ ( 4a2 — x2 ) ; ra a 0 Y1 then**area**ALF ...**find**that DC = 2a √3 = b suppose ; * thus**area**DLK : =**area**CLK = 2a 2a [ dy dx , rb 0 • [ [ " dy dx ; 2a X2 the sum ... Page 118

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**find the area**between the two semicircles OPB and Opb and the straight line bB . B x Let Ob = c , OB = h , then the equation to OPB is r = h cos 0 , and the equation to Opb is rc cos 0. Thus the area π cos e = rde dr . 0 Jc cos e Now Sh ...### Other editions - View all

### Common terms and phrases

Application c² sin² Cambridge circle cloth co-ordinates constant cos² cos³ Crown 8vo curve cycloid definite integral denote differential coefficient double integral dx dx dx dy dz dy dx dz dx element ellipse equal Eulerian integral example expression Fcap Fellow of St Find the area find the volume formula function Hence indefinitely Integral Calculus integrate with respect intrinsic equation John's College latus rectum length limits M.A. Fellow MACMILLAN & CO.'S numerous obtain ordinate parabola partial fractions perpendicular plane positive preceding article radius vector result revolution revolve round round the axis shew sin² solid suppose surface tangent tion tractory transformed integral Trinity College unity vanishes variables vertex x₁ Y₁ πα аф

### Popular passages

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