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 . ; the limit of which, when m and n increase without limit, is the resultobtained. And since every term is of the form axy Ax Ay, we may, asin page 99, call the preceding 2Ax(2axyAy) or 2axy Ax Ay, and itslimit Jdxjaxy dy, fa
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 . ; the limit of which, when m and n increase without limit, is the resultobtained. And since every term is of the form axy Ax Ay, we may, asin page 99, call the preceding 2Ax(2axyAy) or 2axy Ax Ay, and itslimit Jdxjaxy dy, faxy dx dy, or JJ*axy dx dy, if the two operationsare to be represented. And since y is first made variable, we maydenote this by writing dy last of the two, and the symbol of the in-tegral with the limits represented will stand thus : « flf?axydxdy. We may now give a geometrical illustration of the preceding, gene-ralizing the operation into J\f$?* dx dy, where z is a given function. of x and y. Draw the curves y—4>x and y—if/x, and set off theabscissae a and b, OA and OB. Divide the interval AB into anynumber of equal parts m, and having drawn ordinates, divide the partof each ordinate intercepted between the curves into n equal will then be run rectangles, which, as m and n are increasedwithout limit, have for the limit of their sum the area PQRS. This 390 DIFFERENTIAL AND INTEGRAL CALCULUS. limit, compared with the preceding process of summation, will be foundto be represented by fbaf^ dx dy. And this agrees with previousresults; for writing the preceding in the manner first pointed out, wehave Jha dxfff dy, or f\ (tyx—(fix) dx, or J\tyx dx—jba (fix dx, orAQRB —APSB. But if we want to form an idea of the meaning offfz dx dy, we may proceed in either of the following ways. 1. Suppose the area PQRS to be everywhere of different and variablevalue per square unit, in such manner that at the point (x, y) the valueof a square unit, if it were
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