代微积拾级：修订间差异

{{from|積分爲微分之還原，其法之要，在識別微分所由生之函數。如已得 天^二 之微分爲 二天{{RareChar|𢓍|⿰彳天}}，則有 二天{{RareChar|𢓍|⿰彳天}}，即知所由生之函數爲 天^二，而 天^二 即爲積分。<br><br>'''已得微分所由生之函數爲積分。'''而積分或有常數附之，或無常數附之，既不能定，故式中恒附以常數，命爲{{RareChar|𠰳|⿰口丙}}。{{RareChar|𠰳|⿰口丙}}或有同數，或爲〇，須攷題乃知。<br><br>來本之視微分，若函數諸小較之一，諸小較并之，即成函數。故微分之左係一禾字，指欲取諸微分之積分也。如下式 禾二天{{RareChar|𢓍|⿰彳天}}<math>\xlongequal{\qquad} 天^二 \bot</math>{{RareChar|𠰳|⿰口丙}}。來氏說今西國天算家大率不用，而惟用此禾字，取其一覽了然也。|《代微積拾級·卷十七 微積分一·例》}}

{{from|A<span style{{=}}"font-size: 70%;>RTICLE</span> (291.) T<span style{{=}}"font-size: 70%;>HE</span> Integral Calculus is the reverse of the Differential Calculus, its object being to determine the expression or function from which a given differential has been derived.<br><br>Thus we have found that the differential of ''x²'' is ''2xdx'', therefore, if we have given ''2xdx'', we know that it must have been derived from ''x'', or plus a constant term.<br><br>(292.) '''The function from which the given differential has been derived, is called its ''integral''.''' Hence, as we are not certain whether the integral has a constant quantity or not added to it, we annex a constant quantity represented by C, the valueofvalue of which is to be determined from the nature of the problem. <br><br>(293.) Leibnitz considered the differentials of functions asindefinitelyas indefinitely small differences, and the sum of these indefinitelysmallindefinitely small differences he regarded as making up the function; hence the letter S was placed before the differential to show that the sum was to be taken. As it was frequently required to place S before a compound expression, it was elongated into the sign ''∫'', which, being placed before a differential, denotes that its integral is to be taken. Thus,<br><center><math>∫2xdx\int 2xdx = x^2+C</math>.</center>This sign ''∫'' is still retained ever by those whu reject the philosophy of Leibnitz. |''Elements of Analytical Geometry and of The Differential and Integral Calculus'', Integral Calculus, Section I}}