21. Find the area of a loop of the curve r2 = a2 cos 20. Result. a2 22. Find the area contained by all the loops of the curve 23. Find the area between the curves ra cos no and r = a. 24. Find the area of a loop of the curve r2 cos a2 sin 30. Result. = 3a2 a2 4 2 log 2. 25. Find the whole area of the curve r = a (cos 20 + sin 20). 26. Find the area of a loop of the curve (x+y)=4a2x2y2. a2 2 1 2 2 Result. 2π (a+b2). πο +-+). Result. (a2 + b2). 29. Find the area of the loop of the curve y3 — 3axy + x3 = 0. 30. Find the area of the loop of the curve r cos e = a cos 20. Result. (2-7)a2. 32. In a logarithmic spiral find the area between the curve and two radii vectores drawn from the pole. 33. Find the area between the conchoid r = a + b cosec 0 and two radii vectores drawn from the pole. 34. In an ellipse find the area between the curve and two radii vectores drawn from the centre. 35. In a parabola find the area between the curve and two radii vectores drawn from the vertex. 36. Find the area included between the curve r = a (sec + tan 0) and its asymptote r cos 0=2a. Result. (+2) a'. a2. 37. The whole area of the curve ra (2 cos 0+1) is a2 2π + 3/3), and the area of the inner loop is 2 38. Find the whole area of the curve r = a cos 0+ b where a is greater than b. Also find the area of the inner loop. 39. If x and y be the co-ordinates of an equilateral hyperbola x2 - y2 = a2, shew that where u is the area intercepted between the curve, the central radius vector, and the axis. 40. Find the whole area of the curve which is the locus of r2 (a2 + b2) (a2 sin2 0 + b2 cos2 )2 2 41. Find the area included within any arc traced by the extremity of the radius vector of a spiral in a complete revolution, and the straight line joining the extremities of the arc. If, for example, the equation to the spiral bera 2π n , prove that the area corre sponding to any value of 0 greater than 2π is 42. Find the area contained between a parabola, its evolute, and two radii of curvature of the parabola. (Art. 159.) 43. Find the area contained between a cycloid, its evolute, and two radii of curvature of the cycloid. 44. Find the area of the surface generated by the revolution round the axis of x of the curve xy = k3. 45. Also of the curve y = ae* • 46. Also of the catenary y=¦ (e2+e ̄3). 47. Shew that the whole surface of an oblate spheroid is 48. A cycloid revolves round the tangent at the vertex; shew that the whole surface generated is 32 πα. 3 I A cycloid revolves round its base; shew that the whole 50. A cycloid revolves round its axis; shew that the whole surface generated is 8πa (π-). 51. The whole surface generated by the revolution of the tractory round the axis of x is 4πc2. 52. A sphere is pierced perpendicularly to the plane of one of its great circles by two right cylinders, of which the diameters are equal to the radius of the sphere and the axes pass through the middle points of two radii that compose a diameter of this great circle. Find the surface of that portion of the sphere not included within the cylinders. Result. Twice the square of the diameter of the sphere. 53. Find the surface generated by the portion of the curve х y = a + a log between the limits xa and x = ae. a Result. 4a1+√(1+e2)−√2+ log ds 54. Find 1+ √/2 1+√(1+e2), where dS represents an element of surface, and p the perpendicular from the origin upon the tangent plane of the element, the integral being exx2 y2 z2 tended over the whole of the ellipsoid+2+2=1. 178. LET A be a fixed point in a curve APQ, and P any other point on the curve whose co-ordinates are x and y. Let the curve revolve round the axis of x, and let V denote the volume of the solid bounded by the surface generated by the curve and by two planes perpendicular to the axis of x, one through A and the other through P; then (Dif. Cal. Art. 314), dv dx = From the equation to the curve y is a known function of x; suppose (x) to be the integral of πy2; then V = y (x) + C. |