. Differential and integral calculus, an introductory course for colleges and engineering schools. 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. Passing to limits, we have (1) D9V = \p\ 165. Areas: Polar Coordinates. From formula (1) of thepreceding article we get by integration U = iJP*dd + C. Then, by an argume
. Differential and integral calculus, an introductory course for colleges and engineering schools. 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. Passing to limits, we have (1) D9V = \p\ 165. Areas: Polar Coordinates. From formula (1) of thepreceding article we get by integration U = iJP*dd + C. Then, by an argument similarto that of Art. 160, we get (2) u = if6=yP*dd = if6=ylf(e)]2do. Jd=p Example. Let us find the area of a loop of the lemniscate of Bernoulli (Art. 98,example 1) from its polar equation p2 = a2 cos 2 formula (2), the area of the right-hand loop is U = hfJP* do = | JVcob 2 0 tfe - ^(sin 2 8)\ = £ • 166. Exercises (see the curves of Art. 99). 1. Find the area of each loop of the curve p = a sin 3 6. 2. Find the area inclosed by the curve p = 2 a cos 0. 3. Find the area of each loop of the curve p = a sin 2 0. 4. Find the area of the regions inclosed by the curve p = a sin§0. 5. Find the area inclosed by the curve p = a sin3 0. 6. Find the area inclosed by the cardioid p = 2 a(l— cos 0). 236 INTEGRAL CALCULUS 167
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