. Differential and integral calculus, an introductory course for colleges and engineering schools. the entire area between the z-axis and the witch of Agnesi, y = or x = a tan 0, y = a cos2 0. x2 + a2 12. Find the area inclosed by the astroid xl -\- y\ = al, or x = a cos3 d, y = a sin3 0. 13. Find the area of each loop of the curve a*y2 + b2x& = a26V, or x = a sin 6, y = b sin2 0 cos 0. 14. Given the curve (y -2 x2)2 = x5, or x = t2, y = V (t + 2). Find the area of the arch formed with the also the area inclosed by the two branches of the curve and theline x = 4. 15. Find the entir
. Differential and integral calculus, an introductory course for colleges and engineering schools. the entire area between the z-axis and the witch of Agnesi, y = or x = a tan 0, y = a cos2 0. x2 + a2 12. Find the area inclosed by the astroid xl -\- y\ = al, or x = a cos3 d, y = a sin3 0. 13. Find the area of each loop of the curve a*y2 + b2x& = a26V, or x = a sin 6, y = b sin2 0 cos 0. 14. Given the curve (y -2 x2)2 = x5, or x = t2, y = V (t + 2). Find the area of the arch formed with the also the area inclosed by the two branches of the curve and theline x = 4. 15. Find the entire area within the closed curve ay = b2x3 - x4 or ¥ (1 - cos), y 4a (1 — cos ) sin 0. 164. The Derivative of Area: Polar Coordinates. Let U bethe area BOP, bounded by the curve p = f(6), and two of its radii vectores OB and OP. Let p, 6be the coordinates of P. The ra-dius OB is any convenient startingplace from which to measure U. AsP traverses the curve, p turns aboutthe pole, 0, and U plainly varieswith 6. U is, in other words, a func-tion of 8 and has a derivative as. to 6. We seek to determine this 6 take an increment Ad =4- POP. Draw the circular f §§165-166 AREAS 235 arcs PN and PM. Then Ap = NP = PM, and AU = the curvi-linear triangle OPP. It is now evident from the figure that area OPN < area OPP < area OPM,%P2Ad < AU < J(p + AP)2A0, ip2<fd <i(p + Ap)*. orwhence
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
