Exercise · ZW-RIEMANN-INTEGRAL-009 · ★★★ Advanced
A bounded function \(f\) on \([a,b]\) is integrable if and only if…
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- C. for every \(\varepsilon>0\) there is a partition \(P\) with \(U(f,P)-L(f,P)<\varepsilon\)
Explanation
This is Riemann's criterion: the gap between sums can be made arbitrarily small. Continuity is sufficient but not necessary (step functions are integrable).
Review the concept: Riemann integral
A bounded function \(f\) on \([a,b]\) is integrable if the supremum \(L(f)\) of its lower sums and the infimum \(U(f)\) of its upper sums, over partitions \(P\), coincide; that common value is its integral.
\[L(f)=\sup_P L(f,P)=\inf_P U(f,P)=U(f)=\int_a^b f(x)\,dx\]
More practice problems: riemann integral
- ZW-RIEMANN-INTEGRAL-001 Compute ∫_0^1 x dx.
- ZW-RIEMANN-INTEGRAL-002 Compute ∫_0^2 x^2 dx.
- ZW-RIEMANN-INTEGRAL-003 Compute ∫_1^4 5 dx.
- ZW-RIEMANN-INTEGRAL-004 Compute ∫_0^πsin x dx.
- ZW-RIEMANN-INTEGRAL-005 Let f=1 on the rationals and f=0 on the irrationals, over [0,1]. Is it Riemann integrable?
- ZW-RIEMANN-INTEGRAL-006 Which of these functions on [0,1] is not Riemann integrable?
- ZW-RIEMANN-INTEGRAL-007 Let f(x)=0 if x<tfrac12 and f(x)=1 if x≥tfrac12, on [0,1]. Compute ∫_0^1 f.
- ZW-RIEMANN-INTEGRAL-008 Compute ∫_-1^1 x^3 dx.