Again, let area of sector = MENSURATION = r - r x = h, then 19 -r- and r, =*--; therefore " = --(l + l 1 ), and
41 . Area of a Segment of a Circle. (a) When the radius and the angle subtended at the centre are given. In fig. 14, let AEB be a segment of a circle, then its area = sector AC B -triangle ACB. = 4r 2 0-Jr-sine(9, 34) If the segment be greater than a semicircle siuO is negative and the formula becomes as is also geometrically evident. We might in a similar manner find the area of a segment of a circle (j8) when the chord and radius are given, (7) when the chord and its height are given, (5) when the radius and height of the chord are given, () when the chord and angle subtended by the chord are given. In all these cases the method of proceeding is obvious, a segment being the difference between a sector and a triangle. 42. Area of a Lune. Let ADB and ACB (fig. 18) be two seg- inents of circles, then the area of the lune AUBC = segment ADB - segment ACB. Hence if we so choose our data that we can determine the areas of the two segments we have only to take their difference to find the area of the lune. 43. Area of a Circular Zone. Let AB and CD (fig. 19) be two Fig. 18. parallel chords, then the area of the zone ABCD = circle - segment AHB - segment DFC ; or = segment AED + trapezium ABCD + segment BGC = 2 segment AED + trapezium ABCD. 44. The INFINITESIMAL CALCULUS (q. v.) furnishes a simple and elegant proof of the formulae for the areas of a circle and a sector. If y = <p(x) be the equation to a plane curve referred to rectangular axes, then the area between the curve, the axis of x, and two ordinates corresponding to the abscissae a and b is represented by the integral Let x and y be the coordinates of P (fig. 20), then if OP = r the equation to the circle is 2/ a =r 2 -x 2 , and therefore /> . _ area of quadrant AOB =/ Vr 2 - x 2 dx J r 2 . - 1 x x - and therefore area of whole circle = irr 2 . 45. If the equation to a plane curve be given in polar coordi nates, the area bounded by two radii and the curve is equal to where Q l and 2 are the values of 6 corresponding to the limiting radii. For example, let AOP (fig. 20) be 0, then area of circle 2*r 2 - 2 The area of a sector can be found in a similar manner. r- /"Z / VO dv r- as before. B. The Parabola. 46. Length of an Arc of any Plane Curve. If a plane curve be referred to rectangular axes, then the length of any arc of the 1 + *dy 1 + taken between proper limits, i.e., the extremities of the arc. See INFINITESIMAL CALCULUS. 47. Arc of a Parabola. Let the axes of coordinates be the axis ^ of x and the tangent at the vertex A (fig. 21), then, the equation to the parabola being y 2 = 2mx, where m = 2a = latus rectum, we have ?---, and hence ay m s = arc AP = therefore whole arc PAP =-^V?/ 2 + m 2 + mlog,
Since yl = arc PAP = 1> the formula may be written Ty! + - log, f^i 2x t 48. Area of a Parabola. Taking the equation to the parabola in the form y 2 = 4px, we get /"*! area of segment PAP (fig. 21) = 2/ 2/pxdx From these formula; we see that the area of a parabolic segment varies directly as the cube of the square root of the abscissa, and directly as the cube of the ordinate, and that it is equal to | rectangle PQQ P , or triangle PTP . A similar relation holds for the segment cut off by any chord, and thus the area of any parabolic segment can be determined in terms of any data that are sufficient to de termine the segment. 49. Area of a Parabolic Zone. Let PM tftrr 99^ 11 HAT 11 A AT f A NT ** JT ill. ML;. itf //] , V^J-Li i/2> **"! "*!, Aii.1 "^o, and let the ordiuates be inclined to the axis at an angle a. Area of zone PQQ P = segment PAP - segment QAQ Fig. 22. 1 = ipx 1 and yl = kpx v therefore -p 4(x 1 - x. substituting for p we have area cf zone = , and hence on C. The Ellipse. 50. Circumference of an Ellipse. The equation to the ellipse being + = 1 , where a and b are the semiaxes, we have
a 2 o*