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1. Definition of the definite integral

Let a function f defined on a finite interval [a,b]. A partition of [a,b] is a set of intervals: [x₀,x₁], [x₁,x₂],…,[x_n-1,x_n] (1), where a=x₀<x₁<…<x_n=b. (2)

Thus, any set of (n+1) points satisfying (2) defines a partition P of [a,b], which we denote by P={x₀,x₁,…,x_n}. The points x₀,x₁,…,x_n are the partional points of P. the largest of the length of subintervals (1) is the norm of P written as ||P||=max(x_i-x_i-1).

If P and P’ are partition of [a,b] then P’ is refinement of P, if every partition point of P is also a partition point of P’, that is of P’ is obtained by inserting additional points between those of P. If f is defined on [a,b], then a sum δ=Σ f(c_j)(x_j-x_j-1), where x_j-1≤ c_j≤ x_{j, 1}≤j≤n, is a Rieman sum of f over the partition P={x₀,x₁,…,x_n}. Since c_j can be chosen arbitrarly on [x_j; x_j-1], there are infinitely many Rieman sums for a given function f.

Def. Let f be defined on [a,b]. We say that f is Rieman integrable on [a,b] if there is a number L with the following property: for every ε>0, there is a δ>0 such that |δ-L|<ε if δ is any Rieman sum of f over the partition P of [a,b] such ||P||< ε. In this case we say that L is the Rieman integral of f and write

2. The integral as the area under a curve

An important application of the integral is computation of the area bounded by a curve y=f(x), that x – axis and the lines x=a and x=b. For simplicity suppose that f(x)>0 then f(c_j)(x_j-x_j-1) is the area of rectangle with base x_j-x_j-1 and height f(c_j) so the Rieman sum Σ f(c_j)(x_j-x_j-1) can be interpreted as the sum of areas of rectangles.

Th.If f is unbounded on [a,b] then f is not integrable on [a,b].

3. Upper and Lower integrals

Def. If f is bounded on [a,b] and P={x₀,x₁,…,x_n} is a partition of [a,b] let M_j=supf(x), xϵ[x_j-1;x], m_j=inff(x), xϵ[x_j-1;x].

The upper sum of f over P is S(P)=ΣM_j(x_j-x_j-1) and the upper integral of f over [a,b] denoted by f(x)dx – is the infimum of all upper sums.

The lower sum of f over P is S(P)=Σm_j(x_j-x_j-1) and the lower integral of f over [a,b] denoted by f(x)dx – is the supremum of all lower sums.

If m≤f(x)≤M for all xϵ[a,b], then m(b-a)≤S(P)≤S(P)≤M(b-a). Therefore f(x)dx and f(x)dx exist, are unique and satisfy the inequalities: m(b-a)≤f(x)dx≤M(b-a), m(b-a)≤f(x)dx≤M(b-a).

Th. Let f be on [a,b] and let P be a partition of [a,b], then:

(a) – the upper sum of f over P is the supremum of the set of all Rieman sums of f over P;

(b) – the lower sum of f over P is the infimum of the set of all Rieman sums of f over P.

Area A under the curve satisfies the condition: S(P)≤A≤S(P). If f(x)dx=f(x)dx, the f(x)dx exists. If it doesn’t hold, the A is not defined.

4. Existence of the integral

Lemma. Suppose that |f(x)|≤M (1) any xϵ[a,b] and let P’ be a partition of [a,b] obtain by adding r points to a partition P={x₀,x₁,…,x_n}. Then, S(P)≥S(P’)≥S(P)-2M=||P|| (2) and S(P)≤S(P’)≤SP)+2M=||P|| (3)

Proof: suppose that i={1,2,…,n}. If i≠j, the product M_j(x_j-x_j-1) appears in both S(P)=S(P) and cancels out of the difference S(P)-S(P’). Therefore, if M_i1=supf(x)and M_i2=supf(x), then: S(P)-S(P’)= M_i(x_i-x_i-1)- M_i1(c-x_j-1)- M_i2(x_i-c)=±M_ic=c(M_i-M_i1)-x_i-1(M_i-M_i1)+x_i(M_i-M_i2)-c(M_i-M_i2)=(M_i-M_i1)(c-x_i-1)+(M_i-M_i2)(x_i-c) (4)

Since (1) implies that 0≤M_i-M_i1≤2M (r=1,2) (4) implies that 0≤S(P)-S(P’)≤2M(x_i-x_i-1)≤2M||P||.

Suppose that r>1and P’ is obtained by adding point c₁,c₂,…,c_r to P. Let P⁽⁰⁾=P and for j≥1. Let P⁽²⁾ be the partition of [a,b] obtained by adding c_j to P^(j-1). Then the result just proved implies that 0≤S(P^(j-1))-S(^P(j))≤2M||P||, 0≤ j≤ r. Adding these inequalities and taking account of concellations yields 0≤ S(^P(j))-S(P’)≤2M(||P^(j-1)||+ ||P’||+…+||P^(i-1)||. (5) Since P’=P, P^(x)=P’ and ||P^(j)||≤ ||P^(j-1)|| for 1≤k≤k-1 (5) implies that 0≤S(P)-S(P’)≤2Mr||P||, which is equivalent to (2).

Th1. If f is bounded on [a,b], then: f(x)dx≤f(x)dx (6)

Th2. If f is integrable on [a,b], then: f(x)dx=f(x)dx=f(x)dx (7)

Th3. A bounded function f is integrable on [a,b], iff: f(x)dx=f(x)dx (8)

Th4 (fundamental theorem) If f is bounded on [a,b], then f is integrable on [a,b] iff for any ε>0 there is a partition P of [a,b] for which S(P)=S(P)< ε (9)

Th5 (Weierstrass) If f is continuous on [a,b] then f is integrable on [a,b].

Proof: let P={x₀,x₁,…,x_n} be a partition of [a,b]. Since fϵC[a,b], there are points c_j and c_j’ in [x_j-1;x] such that f(c_j)=M_j=supf(x) (10) xϵ[x_j-1;x], and f(c_j)=m_j=inff(x) (11)

Therefore, S(P)-S(P)=Σ(f(c_j)-f(c_j’))(x_j-x_j-1) (12). Since f is uniformly continuous on [a,b], there is any ε>0 and δ>0 such that |f(x’)-f(x)|<ε₁/(b-a)→ M-m< ε₁/(b-a) if x and x’ are in [a,b] and |x-x’|< δ. If ||P||< δ, then |c_j-c_j’|< δ and from (12), S(P)-S(P)< ε₁/(b-a)Σ(x_j-x_j-1)= ε/(b-a)*(b-a)= ε. Hence, f is integrable on [a,b] by fundamental theorem 4.

Th6. If f is monotonic on [a,b], then f is integrable on [a,b].