Theorem:- if F:[a,b]-R is a bounded functions then m(b-a)≤lower R. int. ≤upper R.int.≤ M(b-a).......

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m(b-a)≤ ∫_a^b f(x)dx ≤M(b-a) ; if a≤b || m(b-a)≥ ∫_a^b f(x)dx ≥M(b-a);if b≤a || f ∈ R[a,b] m(b-a)≤ ∫_a^b f(x)dx ≤M(b-a) ; if a≤b || m(b-a)≥ ∫_a^b f(x)dx ≥M(b-a);if b≤a || f ∈ R[a,b]
L13 Real Analysis II If f∈R(α) then m(α(b)−α(a))≤ ∫fdα ≤M(α(b)−α(a)). L13 Real Analysis II If f∈R(α) then m(α(b)−α(a))≤ ∫fdα ≤M(α(b)−α(a)).
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Prove M(b-a) ≤ L(P) ≤ U(P) ≤ M(b-a) || let f be a bounded function on [a,b] #realanalysis #bsc Prove M(b-a) ≤ L(P) ≤ U(P) ≤ M(b-a) || let f be a bounded function on [a,b] #realanalysis #bsc
Theorem I. Let f be a bounded function defined on [a, b] and let m, M be the... Theorem I. Let f be a bounded function defined on [a, b] and let m, M be the...
Riemann Integration | Lecture 16 | Relation between Upper and Lower Sums for Different Partitions Riemann Integration | Lecture 16 | Relation between Upper and Lower Sums for Different Partitions
L16 Real Analysis II If f∈R(α) on [a,b], then cf∈R(α) and ∫cf(x)dα=c∫f(x)dα for any constant c∈R. L16 Real Analysis II If f∈R(α) on [a,b], then cf∈R(α) and ∫cf(x)dα=c∫f(x)dα for any constant c∈R.
L4 Real Analysis II. If bounded function f:[a,b]→R, Lower Reimann Integral ≤ Upper Reimann Integral. L4 Real Analysis II. If bounded function f:[a,b]→R, Lower Reimann Integral ≤ Upper Reimann Integral.