...logarithmic curve, its equation being y = a* f — 1 « = T -- . log. a 13. Prove that the area of the common cycloid is equal to three times the area of the generating circle. 17 31 THE INTEGRAL CALCULUS. 14. ACB is a given semicircle, and DC any ordinate ; bisect the arc AC...

...applying Formula E twice, we shall have versin."1 (',r .-. whole area =3;rr2. That is, the area of the cycloid is equal to three times the area of the generating circle. 211. AREA OP SPIRALS. "We have, by Art. 148, the differential of the area of any segment of a polar...

...is, the area ADE is equal to three times the semi-circle DF'E. Hence the entire area of the (!i GP cycloid is equal to three times the area of the generating circle. (232) Another method of obtaining the area of a cycloid is, to consider that portion of the rectangle...

...that is, the area ADE is equal to three times the semi-circle DF'E. Hence the entire area of the p QP cycloid is equal to three times the area of the generating circle. (232) Another method of obtaining the area of a cycloid is, to consider that portion of the rectangle...

...L, that is the evolute of the cycloid, consists of two half-cycloids as shown in the diagram. 5256 The area of a cycloid is equal to three times the area of the generating circle, and the curve length is four times the diameter of the same circle. PROOF. — (i.) Area PpnN = PprR = BbqQ...

...x = ver-sin— ' ~ — • ydx and the differential equation is dx = 4/2 • _ «3' The area of the cycloid is equal to three times the area of the generating circle. The surface described by the arc of a cycloid when revolved about its base is equal to 64 thirds of...