# First Principles of the Differential and Integral Calculus; Or the Doctrine of Fluxions, Intended as an Introduction to the Physico-Mathematical Scien

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1824 edition. Excerpt: ...the constant C must be zero; that is, we have no constant to add, and we have, generally, the indefinite space APM = \$p?x. But if we wished to estimate the space from the point K, such that AK= b, (b being a known quantity); we should have KPLM=ip x + 6; now this space KPLM becomes zero, when AP or x = b; we have therefore, in that case, 0--p 6 2-f C; therefore C =--f pM, and consequently KPLM = p? x%--\$ps 5. We thus see what purpose is served by the constant, which is added in integrating, and how the conditions of the equation alone can determine it. A 3 i 1 Li We observe that %p x" =%p2 x11 X x; but p x = y; 1 3..1 JL therefore \$p x, or%p xxx=%yx; since therefore 13 p2 x" expresses the space APM, this space will also have for its expression xy, that is, AP X PM, or of the rectangle APMO, whatever AP may be. In like manner p 6 = - / \$ X b; but, when x =AK=b, the equation y = p x gives y--p b, ii _ ii and consequently y = p2 b; that is, KL = p b"; therefore 1 3 11 \$p b or p2 b2 x b =-ST.L X AK; therefore since the space KPLM is represented bypaY--fj? 6, it will have also for its expression f AP x PM--AK x KL, that is, APMO--f AKLL The parabola is the only one of the four conic sections susceptible of being squared. Let us take, as a second example, parabolas of all kinds, whose general equation (30) is +" = am x"; we have m + 2n m + 2n J 1 So that if we wish to estimate the space APM from A, the origin Fig.35. of the abscisses x fig-35), which requires the integral to be zero when APM is zero, and when consequently x is zero, then the constant C is zero, and we have simply m + n APM----xy; that is, the space APM is always a determinate portion of the product xy or of the rectangle...