. Differential and integral calculus. vectorial angle. Geometric Applications 319 CHAPTER VI. GEOMETRIC APPLICATIONS. 205. Definition. The process of determining the area boundedin whole or in part by a curve is termed Quadrature. 206. Quadrature of plane areas. I. When bounded by Algebraic Curves. Let y =f(x) be the equation of any curve as OPC, Fig. 52,and let the area between the curve and the X-axis be generatedby the ordinate (PB) of the curve moving parallel to itself fromleft to right. Let A = areaAPPB, and let PB be anyposition of the generating or-dinate y; then dA, the incre-ment tha
. Differential and integral calculus. vectorial angle. Geometric Applications 319 CHAPTER VI. GEOMETRIC APPLICATIONS. 205. Definition. The process of determining the area boundedin whole or in part by a curve is termed Quadrature. 206. Quadrature of plane areas. I. When bounded by Algebraic Curves. Let y =f(x) be the equation of any curve as OPC, Fig. 52,and let the area between the curve and the X-axis be generatedby the ordinate (PB) of the curve moving parallel to itself fromleft to right. Let A = areaAPPB, and let PB be anyposition of the generating or-dinate y; then dA, the incre-ment that A would take onin any unit of time providedthe change in A becameuniform and so continuedthroughout the unit, is evi-dently the area that PB would describe if its length and velocityremained unaltered throughout the unit. But the velocity ofPB is the same as the rate of change of the distance OB (= x),, it is = dx. Hence the differential area (dA) is a rectanglewhose altitude is y (PB) and whose base is dx (BD), , dA .-. A. Fig. 52. ydx 6 f>* (a) (0 320 Integral Calculus in which b and a (OB and OA) are the limits of integrationtaken along the X-axis. Equation (i) is an expression for thearea bounded by the curve, the Xaxis, and terminal , we find, A=Txdy (2) the expression for the area bounded by the curve, the F-axisand terminal abscissae, b and a being the limits of integrationtaken along trie Kaxis. To illustrate, let it be required to find the area of a parabolicsegment. Here f = 2px. .*. y = \l2px?, hence, = I ydx = fX sJTp&dx =^M = § xy, Jo I , the area of any segment as OBP is § of the rectangle onthe ordinate and abscissa, , \OBPK. II. When bounded by Polar Curves. Let r =/(0) be the equation of any curve as APC, Fig. 53,O being the pole and OX the initial line. Let A = OPP, and let us suppose it to be generatedby the radius vector revolvingaround O as an axis, and chan-ging its length in obedience tothe law expressed in the eq
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