Exercise · ZW-RIEMANN-INTEGRAL-005 · ★★★ Advanced
Let \(f=1\) on the rationals and \(f=0\) on the irrationals, over \([0,1]\). Is it Riemann integrable?
Choose one option.
Show answer
- B. No: the lower integral is \(0\) and the upper is \(1\)
Explanation
Every partition has \(L=0\) and \(U=1\), so the supremum of the lower sums (\(0\)) and the infimum of the upper sums (\(1\)) differ. Being \(0\) except on a countable set is not enough.
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-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.
- ZW-RIEMANN-INTEGRAL-009 A bounded function f on [a,b] is integrable if and only if…