The Integral Calculus

Excerpts

• The Integral Calculus

  We have already shown, that when two functions *increase* or *decrease* without limit, their *ratio* may either increase or decrease without limit, or may tend to some finite limit.

• The Integral Calculus

  Nevertheless the product $\cos\theta × \tan\theta$, of which the first factor diminishes without limit, while the second increases without limit, is always finite, and tends towards the limit $1$; for $\cos\theta × \tan\theta$ is always $\sin\theta$, which last approaches to $1$ as $\theta$ approaches to a right angle, and is $1$ when $\theta$ *is* a right angle.

• The Integral Calculus

  But though the two sums increase without limit when $m$ increases without limit, it does not therefore follow that their ratio increases without limit; indeed we can show that this cannot be the case when all the separate terms of (2) remain finite.

• The Integral Calculus

  If we take any numbers, such as $1$ and $2$, it is evident that between the two we may interpose any number of fractions, however great, either in arithmetical progression, or according to any other law.

• The Integral Calculus

  Generally, if $A$ diminishes without limit at the same time as $B$ increases without limit, the product $AB$ may, and often will, tend towards a finite limit.

• The Integral Calculus

  There is as yet no general agreement on this point of notation.

Equations

• The Integral Calculus

  v = \dfrac{h}{m + 1}

  Sets v equal to the spacing h/(m+1) between successive interposed values of the variable.

• The Integral Calculus

  \frac{(m + 2)A}{(m + 2)a} = \frac{A}{a}

  The ratio (m+2)A over (m+2)a simplifies to A/a, which does not depend on m and is finite.

• The Integral Calculus

  1 + 2 + \dots + (m + 1) = \frac{1}{2}(m + 1)(m + 2)

  The sum of the consecutive whole numbers from 1 to m+1 equals half of (m+1)(m+2).

• The Integral Calculus

  \frac{m + 2}{m + 1}\, ha^{2} + \frac{m + 2}{m + 1}\, ha^{2} + (1 + \alpha)\, \frac{h^{3}}{3}

  The book's expression for the sum after substitution; its second term is printed as ha^2, but the chapter's own expansion gives ah^2 there, so the printed line is an erratum or a transcription error that is flagged here, not corrected.

• The Integral Calculus

  ha^{2} + ha^{2} + \frac{h^{3}}{3} \quad\text{or}\quad \frac{(a + h)^{3} - a^{3}}{3}

  The limit of the sum for x^2 dx is stated as ((a+h)^3 - a^3)/3, but the printed left side has ha^2 twice where the chapter's own expansion gives ah^2, so as printed the two sides do not agree; this is flagged as a possible erratum rather than corrected.

• The Integral Calculus

  \int_{a}^{a+h} x^{2}\, dx

  Defines the integral of x^2 dx between the limits a and a+h as the limit of the sum of x^2 dx over the interval.

Problems

No exercises in this chapter.