. Differential and integral calculus, an introductory course for colleges and engineering schools. ucts of the form f(x)dx could be taken as thedefinition of the definite integral. Indeed, to the originators ofthe Calculus the definite integral first presented itself as the limitof such a sum of products, and from this circumstance arose the name integral and the symbol / ; for, the word integral means the whole regarded as the sum of its parts, and the symbol / is merely an elongated S which originally stood for the word viewing the area under a curve as the limit of a sum of rectan-gl
. Differential and integral calculus, an introductory course for colleges and engineering schools. ucts of the form f(x)dx could be taken as thedefinition of the definite integral. Indeed, to the originators ofthe Calculus the definite integral first presented itself as the limitof such a sum of products, and from this circumstance arose the name integral and the symbol / ; for, the word integral means the whole regarded as the sum of its parts, and the symbol / is merely an elongated S which originally stood for the word viewing the area under a curve as the limit of a sum of rectan-gles, each of the component rectangles is termed an element ofarea. 170. Area of a Sector. In Art. 165 was derived the formulafor the area in polar coordinates of the sector OBC. It was found that area OBC = \ We shall here show how thisformula, may be gotten bymeans of our fundamentaltheorem. We first divide the sectorOBC into n smaller sectors,such as OPP, and then con-struct, from 0 as center,a cor-responding set of n interiorcircular sectors, such as OPN, each having one vertex on the §170 THE DEFINITE INTEGRAL 245 Next we subdivide the small sectors OPP, construct a new setof interior circular sectors, and continue this process indefinitely,taking care that the successive subdivisions shall be so carriedon that each small sector shall have the limit 0. We shall nowshow that area OBC = lim 2) OPN. n=co p To this end we produce the radii vectores beyond the curve, andconstruct from 0 as center a set of n exterior circular sectors suchas OPM. It is geometrically evident that for each value of n, (a) 5) OPN < OBC < 5* OPM. Let $ denote the sum of small curvilinear quadrilaterals suchas PNPM. Then j?OPN -j?OPM=S. Let R be the greatest radius vector in OBC. With R as radius,and center at 0, we strike the circular arc EE = R(y — /3). Thesum of the exterior bases, PM, of the curvilinear quadrilaterals,PNPM, is plainly less than R (7 — /3), and if hn be, for a give
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