Connexion of the Integral with the Differential Calculus

Excerpts

• Connexion of the Integral with the Differential Calculus

  Let $x$ have the successive values $a$, $a + dx$, $a + 2\, dx$, etc., … up to $a + m\, dx$, or $a + h$, $h$ being a given quantity, and $dx$ the $m$th part of $h$, so that as $m$ is increased without limit, $dx$ is diminished without limit.

• Connexion of the Integral with the Differential Calculus

  That is, the integral of $\phi x\, dx$ between the limits $a$ and $a + h$, is $\psi(a + h) - \psi a$, where $\psi x$ is the function, which, when differentiated, gives $\phi x$.

• Connexion of the Integral with the Differential Calculus

  Let us suppose that $\psi a$ is the function of which $\phi a$ is the differential coefficient, that is, that $\psi' a = \phi a$.

• Connexion of the Integral with the Differential Calculus

  which is said to be the integral of $\phi x\, dx$, beginning when $x = a$, the summation being supposed to be continued from $x = a$ until $x$ has the value which it may be convenient to give it.

• Connexion of the Integral with the Differential Calculus

  which is the limit arising from supposing $x$ to increase from $a$ through $a + dx$, $a + 2\, dx$, etc., up to $a + h$, multiplying every value of $\phi x$ so obtained by $dx$, summing the results, and decreasing $dx$ without limit.

• Connexion of the Integral with the Differential Calculus

  It is evident that this series bears a great resemblance to the development in 21, deprived of its first term.

Equations

Problems

No exercises in this chapter.