. Differential and integral calculus, an introductory course for colleges and engineering schools. lines PR and PS. PSPR is the tangent plane to the surface at P, and a study ofthe figure will make it evident that CD =AR = dxz, and DP = BS = dvz,and that consequently dz = dxz + dyz = CP. In words, the total differential is the increment of the z-coordinate ofthe tangent plane. Compare this with the geometric representation of the differ-ential of a function of a single variable given in Art. 73. CHAPTER XXIX MULTIPLE INTEGRALS 213. The Volume under a Surface. Consider the problem offinding the
. Differential and integral calculus, an introductory course for colleges and engineering schools. lines PR and PS. PSPR is the tangent plane to the surface at P, and a study ofthe figure will make it evident that CD =AR = dxz, and DP = BS = dvz,and that consequently dz = dxz + dyz = CP. In words, the total differential is the increment of the z-coordinate ofthe tangent plane. Compare this with the geometric representation of the differ-ential of a function of a single variable given in Art. 73. CHAPTER XXIX MULTIPLE INTEGRALS 213. The Volume under a Surface. Consider the problem offinding the volume of a truncated right cylinder included betweenthe #2/-plane and the surface •z =f(*,y). Let the base of the cylinder be R,whose perimeter C may be a singlecurve or may be made up of partsof several curves. We have alreadylearned that the volume in questionmay be found by cutting the solidinto slices by planes parallel to oneof the coordinate planes and takingthe sum of the volumes of the volume of a slice such as thatshown in the figure may be denoted by A (x) dx, and then. V = fhA(x)dx. To find A(x) requires an integration; for A(x) is the area ofthe plane section AiA2AiA2. The equation of the curve AiA2is z = f(x, y), where x is constant. If we denote the ^/-coordinatesof Ai and A2 by 7 and 5, we have A(x) Jpy=sf(x, y) dy,y=y x remaining constant throughout the integration. 7 and 5 arefunctions of x determined by the equations of (C). Indicating this 321 322 INTEGRAL CALCULUS §214 by writing them y(x), 8(x) and substituting the foregoing valuefor A (x) in the formula for V, we have x=b y = S(x) x=b y=h(x\ V = I I /0, y) dydx = j J zdy dx. x = a y = y(x) x=o y=y(x) Here we have the volume given by two successive the first integration, the inner one, x is to be considered con-stant, and after substituting the limits which are themselves func-tions of x, the resulting function A (x) is to be integrated as to x. By cutting the truncated cylin
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
