. Applied calculus; principles and applications . / pdp. Ap=0 OA JOA Making the summation now with respect to 6, the sum ofthe radial slices is gotten, and the limit of this sum is ^^2 fOA roi roA A = lim y A^ • / pdp= I I pdpdd. Replacing OA and OA by F{e) and / {B) respectively, theformula is I pdedp. (2) When F (d) = 0, the area P1OP2 between the curve p = f (d) PLANE AREAS BY DOUBLE INTEGRATION 309 and the radii is (1), A p2 do, where (2) has been in- tegrated as to p. * If the summing of the elements of area be made first withrespect to 6, keeping Ap constant, RSDP, a segment of acircular
. Applied calculus; principles and applications . / pdp. Ap=0 OA JOA Making the summation now with respect to 6, the sum ofthe radial slices is gotten, and the limit of this sum is ^^2 fOA roi roA A = lim y A^ • / pdp= I I pdpdd. Replacing OA and OA by F{e) and / {B) respectively, theformula is I pdedp. (2) When F (d) = 0, the area P1OP2 between the curve p = f (d) PLANE AREAS BY DOUBLE INTEGRATION 309 and the radii is (1), A p2 do, where (2) has been in- tegrated as to p. * If the summing of the elements of area be made first withrespect to 6, keeping Ap constant, RSDP, a segment of acircular ring, is gotten. A second summation with respectto p gives the sum of such ring segments, the limit of whichsum is the area A. The resulting formula is (p) (p) p dp do, (3) where F-^{p) and/^^p) are the inverse functions of F{d) and/ (d), respectively. Corollary. — dop^A = pdddp and dpe^A = pdpdB are rec-tangles with sides p dS and dp. Example 1. — A simple case of the application of theformulas is in finding the area of the circle p = (2) (3) I pdddp = i / p^ddl 0 t/0 Jo J -j27r = ia^d\ = Tra n27r ra 127r pdpde= I pdpdlJo Jo 310 INTEGRAL CALCULUS In (2) the sectors are summed, while in (3) the rings aresummed. In this case of the circle it is to be noted that no limit need be invoked,since the integral is thesum in each case, the in-crements being the differ-entials, the variables allincreasing uniformly. Example 2. — Find thearea between the two tan-*gent circles p = 2a cos6and p = 2 6 cos ^, wherea>h.
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