. A new treatise on the elements of the differential and integral calculus . x — x^, ?/ z= -|- \^2ax — x\ x=zO, x^:z2a: there-fore, denoting the values of y by — y^ + 2/u / dxdydz 0 *y —y^Jx tan. ^ / _ (tan. d — tan. 6) xdxdy /2« xs/2ax-^xdx0 = (tan. ^ — );ta^.The base of this cylinder is a circle in the plane {x, y) tan-gent to the axis of y at the origin of co-ordinates; and thesecant planes pass through theorigin, and are perpendicularto the plane (2, x). The re-quired volume is therefore theportion of the cylinder includedbetween the sections OP, OF,It can be seen from this exam-ple
. A new treatise on the elements of the differential and integral calculus . x — x^, ?/ z= -|- \^2ax — x\ x=zO, x^:z2a: there-fore, denoting the values of y by — y^ + 2/u / dxdydz 0 *y —y^Jx tan. ^ / _ (tan. d — tan. 6) xdxdy /2« xs/2ax-^xdx0 = (tan. ^ — );ta^.The base of this cylinder is a circle in the plane {x, y) tan-gent to the axis of y at the origin of co-ordinates; and thesecant planes pass through theorigin, and are perpendicularto the plane (2, x). The re-quired volume is therefore theportion of the cylinder includedbetween the sections OP, OF,It can be seen from this exam-ple why, as was observed inArt. 244, Avhen there is a rela-tion between the variables atthe limits of an integral, the order of integration cannot bechanocd without at the same time ascortaininir if it bo notnecessary to make a corresponding change in the limitingvalues of the variables. In this case, after integrating Avithrespect to r,, wo integrate with respect to //. taking the into- gral between the limits y — — {2ax — .r-)*, // .ir -|- ^lax — .r-)*;. 428 INTEGRAL CALCULUS. that is, the integral is considered as bounded by the circum-ference of a circle tangent to the axis of y at the origin; butby what portion of the circumference is not specified until thelimiting values of x are assigned. The integral with respectto X is then taken from ic = 0 to cc == 2a, which thus embracesthe whole circumference. But it is obvious, that, if the order of integration with respectto X and y be reversed, then, that the integral may embracethe whole base of the cylinder, the limits with respect to xmust be xz=a — Va^ — y^, x z=z a-{-Va^ — y^ ] and thosewith respect to y must be y = — a, y — -f- a. We now have,denoting the limiting values o^ xhj Xi^ — x^, /a nx^ n X tan. Q I I dydxdz — a J -x^ J ;rtan. 0 = / j J (^ — )xdydx /« (tan.^- tan.^) Va^ _2/2^^ z=(tan.^ —tan.^)7ta3 (Ex. 2, page 326);which agrees with the first result. 250. IPolar Formul
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