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SOLVED EXAMPLES, at, Example 1. Find L(P, f) and U (P, f) if f (x) =x, for x e10, 3] and let P [0, 1, 2, 3].., be the partition of10, 31, Solution. Let partition P divided the interval [0, 3]into the subinterval I [0, 1],12 =[1, 2], and I3 =[2,3]., Then length of these intervals are given by, (Meerut 2013; Garhwal 2004), d1 =1-0=1, 1

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Riemann ineyrar, 13, Solution. Let the partition P divides the interval [0, 1] into n subinterval such that, 1 2, P =3 0,, r-1 r, n, п п, n n, Clearly, here we have, r-1, , M,, 1, for r=1, 2, ..., n, Mr, and o,, in, Now, by definition, we have, r-1 1, L[P, f]= E m,8,, 1, r= 1, r=1, n n, n r=1, 1, [1+ 2+3+, (п -1).п п-1, 2n2, + (n – 1)]=-, 2, 2n, n, r 1, and, U[P, f]= EM,8, = E, r=1, r=1, и и, 1., 1, [1+2+3+ + n], r=-, 2., n, r=1, n (n+1), 2n2, n+1, 2n, n-1, 1, Txdx = lim L(P, f)= lim, ||P |0, Therefore,, %3D, n → 00 2n, 1, n+1 1, xdx = lim U(P, f)= lim, ||P ||→ 0, and, n 0 2n, 2, From above, it is clear that, -1, x dx, 0-, 3D x dx, Hence,, 2

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12, Integral Calculus, 82 = 2-1=1, 83 = 3-2=1, Let M, and m, be respectively the 1.u.b. and g.l.b. of the function f in [x,y-1 »Xr] then, We, get, M1 =1, m =0, M2 = 2, m2 =1, M 3 =3 and m3 =2, 3, Therefore,, U (P, f)= EM, ô, = M1 d+ M2 82 +M3 83, ア=1, =1.1+2.1+3.1=1+2+3=6, 3, and, L(P, f)= m, 8, = m&1 +m282 +m3 83, r=1, = 0.1+1.1+ 2.130+1+2%= 3., Example 2. Let f (x) = x, 0 <x <1 and lets P= 0, ,,1} be a partition of [0, 1, 42'4, 5-1, find U (P, f) and L(P, f)., Solution. Let the partition P divides the interval [0,1] into the subintervals, 1 3, = 0,, I3, 143D, 24, $4, Clearly, the length of each subinterval is -, 4, Now, let M, and m, respectively be the l.u.b. and g.l.b. of the function fin[x, 1,Xp] then, we get, 1, M3, 2, 1, , m3 =-, 4, 3, ,M2, 4, M4 =1, 4, 1, 3, m4, 4, and, m = 0, m2 =, Therefore,, U (P, f)= M, ò, = M1 d1 +M2 d2 + M3 d3 +M4 84, r=1, 1 1, 1 1, 3 1, - +-,- =]., 4 4, 1, 4 4, 2 4, 4, 1, 1, 1, 16 8, 16 4, 8., 4, and, L[P, f]= m,8, = m,&1 + m2 82 +mz83 + ma84, r=1, 1, 11, 1 1, 3 1, =0- +ー, 4, 4 4, 24 4 4, 1, 1, =0+, 3, 16 8, 16 8, Example 3. Let f (x) = x on [0,1) Find, 1, xdx, and, x dx, by partitioning [0, 1), into nequal parts. Also, show that feR[0, 1]., (Meerut 1991, 92; Garhwal 1997), 3

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14, Integral Ca, n, U[P, f]= EM,8, = 1.8,, 1-ΣΜ3, -ΣL.,, and, %3D, r=1, r=1, = (1-0)=1, Lof= lim L[P, f]=-1, Hence,, n 00, and, o f = lim U[P, f]=L, %3D, n 00, 1, -1, Here, it is clear that S=,, Therefore,, fR [0,1]., Example 5. Find the upper and lower Riemann integrals for the function f defins, [0, 1] as follows :, f (x) = va-x), when x is rational, x is irrational, (Garhwal, (1-x),, when, Solution. We have, (1-x² )-(1-x) =2x (1-x)>0 V xE[0, 1], Therefore, m, = (1-x) and M, =V1-x²., 1, Now,, =1-, 2., = x-, %3D, or, and, f =, -1-, +-sin, 11, -1, sin, %3D, 2 2, 4, -1, 1, Clearly,, 0-, Hence,, f R [0,1], Example 6. Let the function f be defined on 0,, TC, by, cos x, when xis rational, f (x) =-, sin x, when x is irrational, TC, Show that fR 0,, Solution. Let P =:, Hr30,1,...,, be any partition, such that, 4n, 4n, 4, 112

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2, Example 4. Give an example of a bounded function which is not R-integrable over the, interval [0, 1., Solution. Consider the Dirichlet's function given by, (Meerut 1991; Garhwal 1995), when, f(x)={, -D when, x is rational, x is irrational, Clearly, the function f (x) is bounded,, (O í 2, Now, consider a partition P=:, r-1 r, п пп, of the interval [0, 1]. Then we have m, =-1 and M, =1 for all r=1, 2,..., n., n, Therefore,, n, LLP, fl= E m,8, =(-1) 8,, rel (-1). 1 Ax-, ' 33(1-0)%=D-1, r=1., r=1, Lown, 19, r=1 •, R. S'um, rx1 = 1