Exercise · ZW-RIEMANN-INTEGRAL-007 · ★★☆ Intermediate
Let \(f(x)=0\) if \(x<\tfrac12\) and \(f(x)=1\) if \(x\ge\tfrac12\), on \([0,1]\). Compute \(\displaystyle\int_0^1 f\).
Choose one option.
Show answer
- C. \(\tfrac12\)
Explanation
A partition with a tiny subinterval around \(\tfrac12\) makes \(U-L\) as small as desired, and the area under the step is \(1\cdot\tfrac12\).
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.
\[L(f)=\sup_P L(f,P)=\inf_P U(f,P)=U(f)=\int_a^b f(x)\,dx\]
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