Nature of Integration

Excerpts

Nature of Integration
bringing before him modes of speech, which, taken quite literally, are absurd.
Nature of Integration
Hence results a new branch of the inquiry, the reverse of the Differential Calculus, the object of which is, not to find the differential coefficient, having given the function, but to find the function, having given the differential coefficient. This is called the Integral Calculus.
Nature of Integration
For whatever differential coefficient $\psi x$ gives, $C + \psi x$ will give the same, if $C$ be a constant, that is, not varying when $x$ varies.
Nature of Integration
Thus, since $x^{2}$, when differentiated, gives $2x$, $x^{2}$ is the integral of $2x$, beginning at $x = 0$; and $x^{2} - 4$ is the integral beginning at $x = 2$.
Nature of Integration
the sum of an infinite number of infinitely small quantities, which are the differentials or infinitely small increments of a function.
Nature of Integration
so that $mC$, or the sum of all the sides of the polygon, can be made as nearly equal to the circumference as we please.
Nature of Integration
the value of an integral is not to be determined, unless we know the values of $x$ corresponding to the beginning and end of the summation, whose limit furnishes the integral.
Nature of Integration
if $\phi x$ be the differential coefficient of $\psi x$, $\psi x$ might have been called the integral of $\phi x\, dx$.

Nature of Integration
C = -\psi a
The constant C of integration equals minus the value of the antiderivative at the starting point a, so the integral beginning at a is psi x minus psi a.
Nature of Integration
\psi x - \psi a = 0
When x equals the starting value a, the integral psi x minus psi a vanishes, so the integral can be made to begin at a by equating it to zero and solving for x.
Nature of Integration
\psi x - \psi a
The integral of psi' x taken from the starting value a up to x is psi x minus psi a.

No exercises in this chapter.