. Applied calculus; principles and applications . isand the point P moving on the curve according to its equation. By the method of limits, it is evident that, if PR = As, volume MMP > AF > MMP; that is, Triy -\- AyY Ax >AV > wy^ ^x; AV dividmg by Ax, tt (y -\- Ayf > ^ > ^2/^> hence, lim -r— = ^r- = tt/^, since Ay = 0, as Ax = 0;Ax=oL^^J dx .*. dV = iry^dx or V = t f y^ dx. (6) If the revolution is made about a line y = h, then V = 7r f\y-hydx, (8) and when the revolution is about a line x = a, then y = 7r f\x-aydy. (9) Note. — It may be noted that the cone and the sphere
. Applied calculus; principles and applications . isand the point P moving on the curve according to its equation. By the method of limits, it is evident that, if PR = As, volume MMP > AF > MMP; that is, Triy -\- AyY Ax >AV > wy^ ^x; AV dividmg by Ax, tt (y -\- Ayf > ^ > ^2/^> hence, lim -r— = ^r- = tt/^, since Ay = 0, as Ax = 0;Ax=oL^^J dx .*. dV = iry^dx or V = t f y^ dx. (6) If the revolution is made about a line y = h, then V = 7r f\y-hydx, (8) and when the revolution is about a line x = a, then y = 7r f\x-aydy. (9) Note. — It may be noted that the cone and the sphere ofArt. 155 and the paraboloids of Art. 156 and Art. 159 are allsolids of revolution, and hence the formulas of this Art. 160are apphcable to the determination of their surfaces andvolumes. SURFACES AND SOLIDS OF REVOLUTION 291 Example 1. — Find the volume generated by the revolu-tion of the area of the equilateral hyperbola xy = 1 aboutOX. \_Xo Xjx=x, hence, the entire volume has no limit. r 1 l>°=iy = 7r = IT cubic units; \_Xo Xja;=oO. hence the limit of the volume, from the section at Xq = OMq= 1, extending indefinitely to the right, is the same as thevolume of the cylinder generated by the revolution of NPq,the abscissa of Po, about OX. Thus, while the area underthe curve y = 1/x, from the ordinate MqPq at x = 1, in-definitely to the right, is unlimited (as shown in Ex. 11, ), the volume made by its revolution about OX has adefinite limit. According to Art. 156, if the curve y=Tr/x 292 INTEGRAL CALCULUS is drawn, any one of its ordinates in linear units will representthe volume of the solid extending indefinitely to the right ofthat ordinate; thus, in the figure the ordinate MqPq = ttrepresents the volume to the right of PqMqPo, and theordinate MiPi = i tt, the volume to the right of general, the ordinate MP at x = OM represents thevolume of the soHd to the right of the section at any distanceX from the origin, and it represents also the area under
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