Mathematical Methods for Physics and Engineering - Matematica.NET

2.2 INTEGRATIONCombining (2.23) and (2.24) with c set equal to a shows that∫ bf(x) dx = −∫ aabf(x) dx. (2.26)2.2.2 Integration as the inverse of differentiationThe definite integral has been defined as the area under a curve between twofixed limits. Let us now consider the integralF(x) =∫ xaf(u) du (2.27)in which the lower limit a remains fixed but the upper limit x is now variable. Itwill be noticed that this is essentially a restatement of (2.21), but that the variablex in the integrand has been replaced by a new variable u. It is conventional torename the dummy variable in the integrand in this way in order that the samevariable does not appear in both the integrand and the integration limits.It is apparent from (2.27) that F(x) is a continuous function of x, but at firstglance the definition of an integral as the area under a curve does not connect withour assertion that integration is the inverse process to differentiation. However,by considering the integral (2.27) and using the elementary property (2.24), weobtainF(x +∆x) ==∫ x+∆xa∫ xa= F(x)+f(u) duf(u) du +x∫ x+∆xRearranging and dividing through by ∆x yieldsF(x +∆x) − F(x)∆xx= 1∆x∫ x+∆xf(u) du.∫ x+∆xxf(u) duf(u) du.Letting ∆x → 0 and using (2.1) we find that the LHS becomes dF/dx, whereasthe RHS becomes f(x). The latter conclusion follows because when ∆x is smallthe value of the integral on the RHS is approximately f(x)∆x, and in the limit∆x → 0 no approximation is involved. ThusdF(x)= f(x), (2.28)dxor, substituting for F(x) from (2.27),[∫d x]f(u) du = f(x).dxa61