Exercise · ZW-RIEMANN-INTEGRAL-010 · ★★★ Advanced
If \(f\) is integrable on \([0,2]\) and \(1\le f(x)\le3\), what is the tightest interval guaranteed to contain \(\int_0^2 f\)?
Choose one option.
Show answer
- A. \([2,6]\)
Explanation
Every lower sum is \(\ge1\cdot2\) and every upper sum is \(\le3\cdot2\), so \(2\le\int_0^2 f\le6\). Bounds like \([1,3]\) forget to multiply by the length \(b-a=2\).
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.