Question #200746

Define a partition of an interval. Write any three different examples of

partitions of [0,1].

(ii). Let a real valued function 𝑓 be defined and bounded over [π‘Ž, 𝑏]. Prove that 𝑓 is

Riemann integrable over [π‘Ž, 𝑏] if for each πœ– > 0 there is a partition 𝑃 such that

π‘ˆ(𝑃, 𝑓) βˆ’ 𝐿(𝑃, 𝑓) < πœ–

(iii). Show that 𝑓(π‘₯) = π‘₯2

is Riemann integrable over [0,2].


1
Expert's answer
2021-05-31T15:12:00-0400

Ans:-

(i)(i) Partition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set".

three different examples of partitions of [0,1][0,1] are (0,0.4) , (0.4,0.6) , (0.6,1)(0,0.4) \ , \ (0.4,0.6)\ , \ (0.6,1)


(ii)(ii) Let f  be bounded on [a,b][a,b] . Then f  is Riemann integrable if and only if for every  Ο΅ there is a partition on [a,b][a,b]  such that: 0≀U(f,P)βˆ’L(f,P)≀ϡ0≀U(f,P)βˆ’L(f,P)≀ϡ


Since

∫abf=inf[U(f,P)]=sup[L(f,P)]\int_ a ^b f=inf[{U(f,P)}]=sup[{L(f,P)}]


there should exist two partitions P1 and P2P_1 \ and \ P_2  such that:


0≀U(f,P1)βˆ’βˆ«abf<Ο΅20≀U(f,P_1)βˆ’\int _a ^b f<\dfrac{Ο΅}2


and

0β‰€βˆ«abfβˆ’L(f,P2)<Ο΅20≀\int^b_afβˆ’L(f,P_2)<\dfracΟ΅2


The argument is almost identical for L(f,P)L(f,P) (except flipping some inequalities)

Hence the function ff is Riemann integrable


(iii) f(x)=x2(iii) \ f(x)= x^2

For Reimann integrable


β‡’βˆ«abf(x)dx\Rightarrow ∫^b _a f(x)dx

So We have this expression equal to

β‡’βˆ«0bx2dxβˆ’βˆ«0ax2dx\Rightarrow ∫^b _0 x^2dxβˆ’βˆ«^a _0 x^2dx


β‡’(b3βˆ’a3)∫01x2dx\Rightarrow (b^3βˆ’a^3)∫^1_0 x^2dx


Here a=0  and  b=2a=0 \ \ and \ \ b=2

Hence 𝑓(π‘₯)=π‘₯2𝑓(π‘₯) = π‘₯^2 is Riemann integrable over [0,2][0,2]



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