INTEGRAL CALCULUS.

CHAPTER I.

Tur object of the Integral Calculus is to discover the primitive function ima which a given differential co-efficient has been derived.

This primitive function is called the integral of the proposed differential caefficient, and is obtained by the application of the different principies established in finding differential co-efficients and by various transtornations. In veter to avoid the embarrassment which would arise from the perpetual changes of the independent variable, which it would be necessary to effect if we married ourselves to the use of differential co-efficients ainne, we stail genera y aguny differentials according to the infinitesimal method explained in the preceding chapter.

When we wish to indicate that we are to take the integral of a furton we

prefix the symbol f. Thus, if

y = az1

We know that dy = 4er dr

and

If then, the quantity 4ar dz be given in the care of any cruz. A we are desirous to indicate that the primitive fascina ima vaưa i ke ba derived is ar*, we express this by writing

Ssar di = er

When constant quantities are combined with variano que se faHEHE + or — we know that they disappear in taking tam a. formats metr same and therefore they must be restored in taking the morga

Hence in taking the integral of any fazrzna it in pougong them in ad constant quantity, which is usually supersented by the of estime

ve required to find the integral of a quantay men zu

but according to the principles which we have already explained, A is the first differential co-efficient of y or ƒ (r), hence it appears that

If y be a function of x, the first differential co-efficient of y may be cassidered as the ratio of the differentials, or infinitely small increments of y and x; and the differential of y is always equal to the first differential co-efficient of y, multiplied by the differential of x.

In order to find the differential of the product of two variables u and 2, each of which is a function of x, we shall suppose that when z becomes z+d, becomes u + du, and z becomes z + dz.

But du dz being an infinitesimal of the second order, may be neglected.

which agrees with the result already found by a different process.

To find the differential of sin. x according to this method.

Let us now show how we may resolve the problem of tangents by the ne of infinitesimals.

P'Q which is the infinitely small increment of y will be represented

PQ = MM'

The infinitely small triangle P' QP being similar to the triangle PMT, we have,

P'Q : PQ :: PM : MT

dy

The subtangent MT being thus known, we can immediately determine the normal and tangent, and the equation to these lines.

To find the differential of an arc of a curve, we may consider the infinitely small arc PP' included between the ordinates

PM, P'M', as a straight line, and calling the
whole arc of the curve s, the infinitely small
portion PP' will be represented by ds.
The right-angled triangle PPQ gives

T

P

To find the differential of the area comprised between two ordinates PM, P M' of a curve which are infinitely near to each other, neglecting the area PP'Q, if we call the whole area A, the area of the rectangle PM' may be taken for dA

dA = PM × PQ
= ydx

Polar Curces.

In applying the differential calculus to the theory of curves, we have hitherto considered only such as are referred to rectangular co-ordinates. The various propositions which we have demonstrated may, however, be applied to polar curves also, either directly by Taylor's theorem, or, by adapting the expressions already deduced, by aid of the chapters on the transformation of co-ordinates and the change of the independent variable.

The principles of the infinitesimal calculus may also be employed with much elegance in these investigations. Thus, for example,

To find the angle under the radius vector and a tangent at any point of a polar curve.

Let LZ be a curve referred to Polar Co-ordi

mates.

Let S be the pole and SZ the straight line

From which the angles are measured.

Take any point P, and draw a tangent PT.
Let SP=r, angle ZSP = 6.

Take a point Q infinitely near to P, then the re PQ may be considered ultimately as coiniding with tangent and angle PSQ = dễ, raw QR perpendicular to SP, and SY perpenscular on tangent.

2

The following expressions are much employed in the investigations of phys

cal astronomy.

To find the radius and chord of curvature in polar curves.

To find the chord of curvature through S, produce PS and PO to meet the circle of curvature in V and L.

Then since the angle at V is a right angle being in a semicircle, the triangles PVL, PSN are similar.

We shall conclude by showing how the first of the above propositions may be established by the transformation of co-ordinates,

To find the angle under the radius vector and tangent, in a spiral curve.