. An elementary treatise on the differential and integral calculus. (See Mathematical Visitor, 1878, p. 26.) 208. Mixed System of Co-ordinates.—Instead of dividing a solid into columns standing on rectangular bases, so that z dx dy is the volume of the infinitesimal column, it is sometimes more convenient to divide it into infinitesimal columns standing on the polar element of area abed = r dr dd, in which case the corresponding parallelopipedon is represented by zr dr dd, and the expression for V becomes. Fig. 59. V = J Jzr dr dO, taken between proper limits. From the equation of thesurface,
. An elementary treatise on the differential and integral calculus. (See Mathematical Visitor, 1878, p. 26.) 208. Mixed System of Co-ordinates.—Instead of dividing a solid into columns standing on rectangular bases, so that z dx dy is the volume of the infinitesimal column, it is sometimes more convenient to divide it into infinitesimal columns standing on the polar element of area abed = r dr dd, in which case the corresponding parallelopipedon is represented by zr dr dd, and the expression for V becomes. Fig. 59. V = J Jzr dr dO, taken between proper limits. From the equation of thesurface, z must be expressed as a function of r and 6. EXAMPLES 1. Find the volume included between the plane z = 0,and the surfaces x2 + y2 = \.az and y2 = 2cx — x2. Here z = —-—— = —; hence the z-limits are —- and 4za Aa The equation of the circle y2 = 2cx — x2, in polar co-or-dinates, is r = 2c cos 6; hence the r-limits are 0 and 2c cos 6, or 0 and rx; and the 0-limits are 0 and 2 V=2 i-dBdr Jo Jq 4=a 2c* 2c* /*** a Jq 377C4 8 a cos4 6 dd 2c4 Att. (Ex. 4, Art. 157.) EXAMPLES. 389 2. The axis of a right circular cylinder of radius b,passes through the centre of a sphere of radius a, whena > b; find the volume of the solid common to bothsurfaces.* Take the centre of the sphere as origin, and the axis ofthe cylinder as the axis of z; then the equations ol thesurfaces are x2 + y2 + z2 = a2 and x2 + y2 = b2; 01% interms of polar co-ordinates, the equation of the cylinderis r = b. Hence f
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