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.

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L18 Real Analysis II If f∈R(α) on [a,b], then ∣f∣∈R(α) on [a,b] and | ​∫fdα​| = ∫ ∣f∣ dα.

L14 Real Analysis II If f ∈R(α) & f(x)≥0 then ∫f(x)dα(x)≥0.If f,g ∈R(α)& f(x)≥g(x) then ∫fdα ≥ ∫gdα.

L20 Real Analysis II If f∈R(α), then f∈R(cα). Also ∫ f(x)d(cα(x))=c∫f(x)dα(x).

f∈R([a,b])⇒cf∈R([a,b]) and ∫(cf)(x)dx=c∫f(x)dx

L15 RA II If f,g∈R(α) & f(x)≤g(x) then ∫fdα ≤ ∫gdα. If f∈R(α) & ∣f(x)∣≤k then |∫fdα| ≤k(α(b)−α(a)).

(M.sc-Ⅰ)If f ∈R(α) than cf∈R(α) where c is any real number also show that ∫cf d(α)=c∫f d(α)

L17 Real Analysis II If f∈R(α1) and f∈R(α2) then f∈R(α1+α2), and ∫fd(α1+α2) = ∫fdα1 + ∫fdα2.

Find the upper and lower R-integrals of a function fin [0,2] where f(x) equal. #addmaths #alevelmath