The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . han the area MPP M, the value of which it is our object to investi-gate, by the sum of the curvilinear trian-gles Vrp, prp, prp, and pRP. Thesum of these triangles is less than thesum of the parallelograms Qr, qr, qr,and gR ; bu
The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . han the area MPP M, the value of which it is our object to investi-gate, by the sum of the curvilinear trian-gles Vrp, prp, prp, and pRP. Thesum of these triangles is less than thesum of the parallelograms Qr, qr, qr,and gR ; but these parallelograms aretogether equal to the parallelogramqw, as appears by inspection of thefigure, since the base of each of theabovementioned parallelograms is equalto mM., or qV, and the altitude Pwis equal to the sum of the altitudes ofthe same parallelograms. Hence thesum of the parallelograms Mr, mr, mr, and mR, differs from the cur-vilinear area MP PM by less than the parallelogram qw. But this lastparallelogram may be made as small as we please by sufficiently increas-ing the number of parts into which MM is divided ; for since one side ofit, Pw, is always less than PM, and the other side Pg, or mMf, is assmall i part as we please of MM, the number of square units in qw, isthe product of the number of linear units in Pw and Pg, the first of Fig. THE DIFFERENTIAL AND INTEGRAL CALCULUS. 61 which numbers being finite, and the second as small as we please, theproduct is as small as we please. Hence the curvilinear area MP PMis the limit towards which we continually approach, but which we neverreach, by dividing M M into a greater and greater number of equal parts,and adding the parallelograms Mr, rar, &c, so obtained. If each of theequal parts into which MM is divided be called dx, we have OM = a,Om = a -j- dx, Om ~ a -J- 2dx, &c. And MP, mp, mp, &c, are thevalues of the function which expresses the ordinates, corresponding to a,a -j- dx, a + 2dx, &c, and may there
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