. Differential and integral calculus. face is cylindrical, dS = Pds. (a) (1) 334 Integral Calculus Similarly, dV = volume ASpSAI/, = Adx. .. V = JAdx, in which A = area SDS and dx = SA. (2) Formulae (i) and (2) are obviously applicable to all caseswhere P can be expressed in terms of s and where A can beexpressed as a function of x. Cor. If SpS is a circle with its center D in the X-axis; then P = 2 iry, A = Try2, and (1) and (2) become, respectively, S = 2 I 7T>7ft as heretofore determined. See § 209. EXAMPLES. 1. To find the surface and volume of a regular pyramid orcone. 1. To
. Differential and integral calculus. face is cylindrical, dS = Pds. (a) (1) 334 Integral Calculus Similarly, dV = volume ASpSAI/, = Adx. .. V = JAdx, in which A = area SDS and dx = SA. (2) Formulae (i) and (2) are obviously applicable to all caseswhere P can be expressed in terms of s and where A can beexpressed as a function of x. Cor. If SpS is a circle with its center D in the X-axis; then P = 2 iry, A = Try2, and (1) and (2) become, respectively, S = 2 I 7T>7ft as heretofore determined. See § 209. EXAMPLES. 1. To find the surface and volume of a regular pyramid orcone. 1. To find the surface. Let P = perimeter of base and Oc = h — slant mnd be the position of the generating perimeter P atany instant. Since P and P are similar, we haveP _ Od _ s m p7~~o^~J,; Phence P= -=y s. n This value of Pva (1), § 210, gives P S ti Q PhS= — p nhf TjBsds> Geometric Applications 335 Hence the convex surface of any pyramid or cone is measuredby i product of perimeter of its base by its slant Fig. 57-2. To find the volume. Let Ar = area of base and Oa =then h = altitude. Let Ob = x; AAf OFOa Hence, §210, (2), A-$*. _A^ Ch~ & J 0 x2dx, V = Ah , the volume of any cone or pyramid is measured by \ ofthe product of its base and altitude. 2. Show that the volume of the frustum of any pyramid or cone is equal to - (A + A + \lAA) where A and A are thebases, and h is its height. 3. Find the volume of a right conoid with circular base, the radius of base being a, and altitude h. Ans. ?no?h 4. Find the volume of the wedges cut from a tree (radius =a) in cutting it down, the faces of each wedge being inclined atan angle of 450. Ans. ±as. 336 Integral Calculus CHAPTER VII. SUCCESSIVE INTEGRATION. 211. Successive integration is a reversal of the process ofsuccessive differentiation. If, for example, x is equicrescentand dsy = xdx3, then, / d3y — dx2 I x?dx; , d*y = ds?S- + C Again, I d2y = dx I \ h Cdx ?; , dy = dx \ f- Cx + Q > •
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