. Differential and integral calculus. 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
. Differential and integral calculus. 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 equa-tion r *=/(&). Obviously the rateof change of A, , dA, is thecircular sector OPB describedby OP in any unit of time with a constant angular velocity (d$). Hence, since BP = rdd and OP = r, we have. Geometric Applications 321 dA = \ = \r2dQ\ , (b) .-. a=* r w* (3) where <£ = ^tfJT and ^ = POX. Equation (3), it will be ob-served, gives the area bounded by the curve and terminal radii-vectores. For example, let us find the area of one loop of the lemniscata, r2 = a2 cos 2 0.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
