. An elementary treatise on the differential and integral calculus. V%ry — y2 |7rr2 (See Ex. 6, Art. 151) = i the area of the cycloid. Sinceintegrating between the limits includes half the area ofthe figure. * The student should pay close attention in every case to the limits of theintegration, AREA BETWEEN PARABOLA AXB CIRCLE. 363 Therefore the whole area = 3t/2, or three times the areaof the generating circle.* 189. The Ellipse.—The equation of the ellipse referredto its centre as origin, is ay + Vlxl — <ffi; therefore the area of a quadrant is represented by aJn v J dx = - ~- (See Art. 1
. An elementary treatise on the differential and integral calculus. V%ry — y2 |7rr2 (See Ex. 6, Art. 151) = i the area of the cycloid. Sinceintegrating between the limits includes half the area ofthe figure. * The student should pay close attention in every case to the limits of theintegration, AREA BETWEEN PARABOLA AXB CIRCLE. 363 Therefore the whole area = 3t/2, or three times the areaof the generating circle.* 189. The Ellipse.—The equation of the ellipse referredto its centre as origin, is ay + Vlxl — <ffi; therefore the area of a quadrant is represented by aJn v J dx = - ~- (See Art. 186) = iabn. Therefore the area of the entire ellipse is ~ab. 190. The Area between the Parabola y2 — axand the Circle y2 = 2a x — x2. — These curves passthrough the origin, and also intersect atthe points A and B, whose common abscissais a. Hence, to find the area includedoetween the two curves on the positive side 0of the axis of x, we must integrate betweenthe limits x = 0 and x = a. Therefore,bv Art. 185, we have A = fa[(2ax - z2)i- - (ax)] cL«-o. Fig. 53. |-rt8; (See Ex. 6, Art. 151.) which is the area of OPAP. * This quadrature was first discovered by Roberval, one of the most distin-guished geometers of his day. Galileo, having failed in obtaining the quadratureby geometric methods, attempted to solve the problem by weighing the area of thecurve against that of the generating circle, and arrived at the conclusion that theformer area was nearly, but not exactly, three times the latter. About 162S,Roberval attacked it. but failed to solve it. After studying the ancient Geometryfor six years, he renewed the attack and effected a solution in 1634. (See SalmonsHigher Plane Curves, p. 266.) 364 THE SPIRAL OF ARCHIMEDES.
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