Syllabus
· Lower
Darboux sum and upper Darboux sum and related results
· Darboux
theorem
· Conditions
of Riemann integrability
· Integrability
criterion in terms of Riemann sums
· Properties
of Riemann integral
· Properties
of integrals
· Bonnet's
mean value theorem
· Indefinite
integral theorem
· Second
fundamental theorem of integral calculus
· Integration
by parts
· Change of variable
Partition
By a partition P of a closed interval [a, b], we mean a finite ordered set
P = {a = x0, x1, …, xn = b}, where a = x0 < x1 < x2 < … < xn = b.
The partition P consist (n + 1) points which divide the closed interval [a, b] into n sub-intervals [x0, x1], [x1, x2], …., [xr–1, xr], …, [xn –1, xn]. We shall use the symbol dr to denote the rth sub-interval [xr–1, xr] and its length (xr – xr –1). When function f is bounded on [a, b], then the symbols Mr and mr are used to denote the supremum and infimum of f on rth subinterval dr.
Family of Partition
The collection of all partitions of closed interval [a, b] is denoted by P [a, b] is known as the family of partitions of [a, b].
Norm
The norm of a partition P is denoted by ||P|| or m(P), will mean the length of the largest subinterval dr. Thus, m(P) = max {xr – xr–1: 1 £ r £ n}.
Finer
If P1 and P2 are the partitions of [a, b], i.e. P1, P2 Î P [a, b], then the partition P2 is said to be finer than P1 if P1 Ì P2, i.e. every points of partition P1 is the point of P2.
Upper and lower Darboux sums
Let f be a bounded function on [a, b] and Mr and mr respectively denote its upper and lower bounds on dr. Then the sums U(P) and L(P) defined by
Note: The upper and lower Darboux sums of a function f may also be denoted by U(P, f ) and L(P, f ) respectively. It should be note that
U(P, –f) = – L(P, f ) and L(P,–f ) = – U(P, f ).
Q.1.
State
and prove the four important properties regarding the upper and lower Darboux
sums for any function f bounded on [a, b].
Soln: The following are the four important properties regarding the upper and lower Darboux sums for any function f bounded on [a, b]:
i. If PÎP [a, b], then U(P) and L(P) are bounded and U(P) ³ L(P) "PÎP [a, b]
ii. If P1, P2 Î P[a, b] and P1 Ì P2, then U(P2) £ U(P1) and L(P2) ³ L(P1), i.e. the upper and
lower Darboux sums can not increase and decrease respectively with the fineness of the partition.
iii. L(P1) £ U(P2) " P1, P2 Î P[a, b], i.e. no lower Darboux sum can exceed any upper Darboux sum.
iv. If P1, P2 Î P[a, b], || P1||£d and |f(x)| < k " x Î [a, b] then for P1 Ì P2 with p additional points U(P1) £ U(P2) + 2pkd, and L(P2) £ L(P1) + 2pkd
= Mr(xr – x1 + x1 – xr –1) – Mr(x1– xr –1) – Mr(xr–x1)
= Mr (xr – x1) + Mr (x1 – xr –1) – Mr(x1 –xr –1) – Mr(xr – x1)
= Mr(xr – x1) + Mr (x1 – xr –1)– Mr(x1– xr –1) – Mr(xr – x1)
= (Mr – Mr) (x1 – xr –1) + (Mr – Mr) (xr – x1)
Since, |f(x)| < k "
x Î [a, b]
– k £ Mr£ Mr £ k and – k £ Mr £ Mr £ k
Þ 0 £ Mr – Mr£ 2k and 0 £ Mr – Mr£ 2k
Therefore, U (P1) – U (P2¢) £ 2k (xr – xr – 1) £ 2kd, || P1|| £ d.
Thus U(P1) £ U(P¢2) + 2kd
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