. A new treatise on the elements of the differential and integral calculus . 402 INTEGRAL CALCULUS. the second revolution, the radius vector again describes thisarea, and also the area PBAB included between the first andsecond spires. Hence the area PBAB is measured by It is evident that during any, as the m*^, revolution, theradius vector describes the whole area out to the m*^ spire,and that, to find this area, the integral A J O must be taken between the limits (m — 1)2;? and 2m;r, whichwill give for this area denoted by u 6 ^a\27ty^m-{m — iyL In like manner, we have for the entire area den
. A new treatise on the elements of the differential and integral calculus . 402 INTEGRAL CALCULUS. the second revolution, the radius vector again describes thisarea, and also the area PBAB included between the first andsecond spires. Hence the area PBAB is measured by It is evident that during any, as the m*^, revolution, theradius vector describes the whole area out to the m*^ spire,and that, to find this area, the integral A J O must be taken between the limits (m — 1)2;? and 2m;r, whichwill give for this area denoted by u 6 ^a\27ty^m-{m — iyL In like manner, we have for the entire area denoted by u^out to the (m — 1)*^ spire, u = ^a(27ty I (m - ly - (m - 2y]: u - u = ^a\27ty^m — 2 {m- ly + {m - 2y]^, which is the expression for the area included between the (m — 1)*^ and the m^^ spires. If we suppose a = ^ , this formula becomes u—u==~27t\m-2(m-iy + {m-2y\ m-2(m-iy-\-(m — 2y . ,,^ o and in this, making w = 2, we find 27t for the area includedbetween the 1^* and 2*^ spires. Hence the area included be-. QUADRATURE OF CURVES. 403 tween the {m — 1)*^ and m*^ spires is m — 1 times thatincluded between the 1^* and 2^ spires. 239, The quadrature of curvilinear areas is sometimesfacilitated by transforming rectilinear into polar co-ordinates. Take, for example, the folium of Descartes^ which, referredto rectangular axes, is represent-ed by the equation x^ -\- y^ — f^^y = 0. This curve is composed of twobranches, infinite in extent, whichintersect at the origin of co-ordi-nates, and which have for a com-mon asymptote the straight line of which the equation is To determine the area of any portion of this curve in termsof the primitive co-ordinates, we must find what the integral ofydx becomes when in it the value of y derived from the equa-tion of the curve is substituted. This requires the solution ofan equation of the third degree ; but if rectilinear be changedinto polar co-ordinates, the pole being at 0, there will be butone value of the radius vec
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