This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ...of solids of revolution. A plane area AEFIi, Fig. 68, bounded by the curve EF, whose equation is y = f(x), the axis of.Y and the ordinales and RF, rotates about the axis OX and thereby generates a solid of revolution. To derive an expression for the volume of this solid we proceed as follows: Let the interval AB be ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ...of solids of revolution. A plane area AEFIi, Fig. 68, bounded by the curve EF, whose equation is y = f(x), the axis of.Y and the ordinales and RF, rotates about the axis OX and thereby generates a solid of revolution. To derive an expression for the volume of this solid we proceed as follows: Let the interval AB be divided into n parts, A-r1(:2, Ajs, etc., and lei ordinales be erected at the points of division. In any subinterval Aj, let yi and -//' be respectively the smallest and largest numerical values of the ordinale, and construct reclangles having Axj as a base and?//and y," respeclivejy, as altitudes. Repeating this construction for each of the subintervals we obtain one plane area made up of rectangles lying entirely below the given curve EF and u second plane area made up of reclangles whose upper bases lie above Ihis curve. The solids obtained by revolving these areas about the A'-axis have respectively the volumes b V" = X-y"z x. The volume V of the solid generated by the revolution o the area AEFB must lie between V and V." That is, 22/'2A.r V 2xy"Sx. (1) a a It will be seen that V and V" are both monotone functions, the first never decreasing and the second never increasing; hence since the functions are finite, each has a limit as x = 0 (Art. 12). Following the methed of Art. 80, it may be shown that the two limits are equal. Consequently we may write i-b V = L T 2/'2A.r = L y. ny"2 x. (2) Since y is taken to be considered a single-valued and continuous function of a:, we may replace the common limit by the definite integral and write V dx. (3) By a similar process it can be shown that the volume of the solid generated by a rotation about the F-axi...
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