Hence the area swept out by r during one revolution of the generating circle is 4 a Subtracting from circle which is ab, A (3a + 2b). a the epicycloid and the fixed circle When a curve forms a loop, the area may sometimes be conveniently found by taking y or the tangent of the angle 00 which the radius makes with the axis of r, as the independent y A = 1⁄2 fr2 d0 = 1 fd0a2 sec2 0 = [dtx2. (21) The curve y3-3axy + x3 = 0, has a loop which touches the axes of x and y at the origin; and taking this from t0 to t = ∞ we have A = 3a2 for the whole area of the loop. (22) The lemniscate whose equation is (x2 + y2)2 = a2x2 — b3y3, has two loops; find its area. a2 - b2 t2 A = 1⁄2 [dt x2 = 1 [dt (1 + 12)? a b and for each loop the limits of t are and SECT. 2. Rectification of Curves. When a curve is referred to rectangular co-ordinates, the length of any portion of it is found by integrating between the proper limits. (1) The equation to the common parabola being the length of an arc measured from the vertex is The length of the are measured from the origin is This was the first curve which was rectified. The author was William Ncil, who was led to the discovery by a remark of Wallis in his Arithmetica Infinitorum. See Wallisii Opera, Tom. 1. p. 551. (3) To find when the curves expressed by the equation Hence the whole length of the cycloid is 8a or four times the diameter of the generating circle. discovered by Wren. This rectification was (5) The equation to one of the hypocycloids is x3 + y3 = a3; the whole length of the curve is 6a. (6) The equation to the catenary is the arc being measured from the point where y = c. Hence as the area is equal to c (y2 – c2), it is equal to cs; that is, the area contained between the axes, the curve and any ordinate is equal to the length of the corresponding arc multiplied by a constant. (7) The equation to the tractory is Then dy dx (a2 — y3)3 supposing the arc to be measured from the point where y = a. When a curve is the evolute of another curve, the length of its arc is best found by taking the difference of the radii of curvature of the involute, corresponding to the extremities of the arc. (8) To find the length of the evolute of the ellipse. The radius of curvature of the ellipse at the extremity of the b2 major axis is; that at the extremity of the minor axis is a2 a therefore the length of the fourth part of the evolute is b2 α = 13 b a2 b If the curve be referred to polar co-ordinates r and 0, supposing it to be measured from the pole. Hence the arc is equal to the portion of the tangent at its extremity, which is intercepted between the point of contact and the subtangent. (10) The equation to the spiral of Archimedes being log This is the same as the arc of a parabola (whose latus rectum is 2a) intercepted between the vertex and an ordinate equal to r. (11) In the involute of the circle the equation to which is r2 — p2 = a2, If the arc be measured from the point where r = a we length of the arc of the circle which is unwound, so that if this be called a0, where ca+2b, a and b being the radii of the fixed and generating circles respectively. Hence b The corresponding arc in the hypocycloid is 8 (a - b). For a curve of double curvature we have 2 • - ¡da {1 + (da)2+ (da) '}'. = (13) In the helix, therefore dx dy y = a cos nx, ≈ = a sin nx; - a s = fdx (1 + n2 a2)3 = (1 + n2a2)3 x + C. If the arc be measured from the origin C = 0, and 8 = (1 + n2 a2) 3 x. |