Some measures of areas and lengths

Areas and lengths for special cases

Measure of lengths

To obtain the measure of the length for a line segment, straight or not, it is appropriate to use integration. It can be used a formula for lines represented by continuous functions.

If  y = f(x) is a continuous function in the closed interval [a, b], and there is a continuous derivative in that interval, then the measure of the length for the line segment represented by the above function for x in [a, b] is the real number s given by

This formula is based in calculus themes like the intermediate value theorem and Rolle's theorem. Essentially, this formula comes from inscribing polygonals in the considered curved line. The sum of the measure of lengths of the polygonal sides approximates the  looking for lenght. Such approximation will be better for poligonals better adjusted to the considered curved line. The number of sides for the inscribed polygonal increases as the length of each side decreases.

Measure of the length for a straight line segment

To apply above formula in the case of a straight line segment, let consider the segment with end points P, Q, in the straight line represented by f(x) = mx + h.

Then P = (a, f(a)), Q = (b, f(b)). The distance formula used in coordinate geometry gives,

That is,

in the situation of next illustration where b is greater than a.

To apply the integration formula, if  f(x) = mx + h, then f ‘(x) = m.

Taking the established limits of the integral, PQ is (as before)

If PQ is parallel to x axis, then m = 0 and the result is  PQ =b – a.

Measure of the length for a segment of a parabola

For a parabola whose focus is F = (0, p) the corresponding function is f(x) = (1/4p)x², for x in [0, 2p], for instance.

Then  f ‘(x) = x/2p and 1 + [f ‘(x)]² = (4p² + x²)/4p²

To find the measure of the length of the considered arc, calculate:

Using a table of integrals or a program ( like Mathematica),

Where K is an indeterminate real number which appears with every indefinite integral.
If limits of integration ( 2p, 0) are applied,

For example, if  p = 2, 4.591 is an approximation for the measure of the length of the considered arc, in the used lenght units (inches, centimeters, ...)

For a more general situation, if x varies in the interval [0, c], where c is a positive real number,

 

If the focus of the parabola is  (0, 2) and c = ½ then s ≈ 2.0052 in the used lenght units.

Measure of the length for a circular arc

For a circle represented by  x² + y² = r², it will be considered  the function

 defined for x in [0, r]  that is the arc included in the first quadrant.

then,

It has to be done

If x = r sin t, then dx = r cos t dt, and r² – x² = cos² t.

If x = 0, r sen t = 0 and t = 0.

If x = r,  r = r sin t, that is, 1 = sin t and t = π/2.

The integral to be calculated is:

The result of integration is r.t, with the given limits,  r(π/2 - 0) = (π/2)r.

The measure for the circle is 4 times the former result, that is 2πr.

Measure of the length for a segment of an ellipse

Given the ellipse represented by  (x/a)² + (y/b)² = 1, it can be done a process similar to that of  a circle and consider the following function for x in [0, a], corresponding to the part of the curve in the first quadrant.

In this way,

The integral for this function of x is one of the so called elliptic integrals, which are not calculated by conventional means. It has to be used infinite series or a computer program like the above mentioned to obtain for this case,

A similar situation happens for an arc of hyperbola

 A una situación similar se llega si se quiere calcular la medida de la longitud de un arco de hipérbola.

Measure of the area for a plane elliptic region

An ellipse with coordinate axes as symetry axes is represented on the cartesian plane by

 (x/a)² + (y/b)² = 1,

from which it is obtained, 

The former equation includes two functions which represent, respectively, the line in the half-plane for which y ≥ 0 and the one for which y ≤  0.

Because of the symetry of the curve, it is necessary to determine the measure of the area for the convex shaded region and take it 4 times.

Using integral calculus,

The result can be obtained by means of an adequate computer program:

In the following schema,

Then it can be written,  x = a sen t,  c = a cos t, dx = a cos t dt.

but cos² t = (1 + cos 2t)/2.

Then,

 Now, for

2t = u  and  du = 2dt, dt = (1/2)du.

Then,

and

where  K₂ = (a²/2)K₁; K₁ is an arbitrary constant

From above schema,

 t = arcsin (x/a)  and sin 2t = 2( sen t)(cos t) = 2(x/a)(c/a) =  2xc/a²

Now,

K  is an arbitrary constant.

Finally, for the definite integral, x takes the value of a:

(a²/2)arcsin (1) + (1/2)a.0  = (a²/2)(π/2) = πa²/4. If x takes the value of 0, the result is 0.  Then,

 The measure of the total elliptic region is πab expressed in area units.