Louisiana State University



Section 5.2 Definite IntegralsTopic 1: Net AreaWhen the graph of a function lies below the x-axis for some interval [a,b], the Riemann sum on that interval will be negative. When this is the case, the Riemann sum approximates the negative of the area of the region bounded between the curve and the x-axis.DefinitionNet AreaConsider the region R bounded by the graph of a continuous function f and the x-axis between x=a and x=b. The net area of R is the sum of the areas of the parts of R that lie above the x-axis minus the sum of the areas of the parts of R that lie below the x-axis on [a,b].Topic 2: The Definite IntegralThe Riemann sums used so far involve regular partitions in which the subintervals have the same length ?x. We will now look at partitions of [a,b] called general partitions in which the lengths of the subintervals are not necessarily equal. The general partition is used to define the general Riemann Sum. DefinitionGeneral Riemann SumSuppose x0,x1, x1,x2,…,xn-1,xn are subintervals of [a,b] witha=x0<x1<x2<…<xn-1<xn=b.Let ?xk be the length of subinterval xk-1,xk and let xk* be any point in xk-1,xk, for k=1,2,…,n.If f is defined on [a,b], the sumk=1nfxk*?xk=fx1*?x1+fx2*?x2+…+fxn*?xnis called a general Riemann sum for f on [a,b]. We consider the partitions in which we let ? denote the largest value of ?xk; that is, ?=max?{?x1, ?x2, …,?xn}. Observe that if ?→0, then ?xk→0, for k=1,2,…,n. Forlim?→0k=1nfxk*?xkto exist, it must have the same value over all general partitions of [a,b] and for all choices of xk* on a partition. DefinitionDefinite IntegralA function f defined on [a,b] is integrable on [a,b] iflim?→0k=1nfxk*?xkexists. This limit is the definite integral of f from a to b.abfx?dx=lim?→0k=1nfxk*?xk Topic 3: Evaluating Definite IntegralsTheoremIntegrable FunctionIf f is continuous on [a,b] or bounded on [a,b] with a finite number of discontinuities, then f is integrable on [a,b].Topic 4: Properties of Definite IntegralsProperties of Definite IntegralsLet f and g be integrable functions on an interval that contains a, b, and p.aafx?dx=0 (definition)abfx?dx=-bafx?dx (definition)abfx+gx?dx=abfx?dx+abgx?dxabcfx?dx=cabfx?dx for any constant cabfx?dx=apfx?dx+pbfx?dxTopic 5: Evaluating Definite Integrals Using LimitsIf f is integrable on [a,b], thenabfx?dx=lim?→0k=1nfxk*?xkfor any partition of [a,b] and any point xk*. To simplify the calculations, we will use equally spaced grid points and a right Riemann sum to evaluate definite integrals using limits. abfx?dx=lim?→0k=1nfxk*?xk=limn→∞k=1nfa+k?x?x ................
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