Exercise · ZW-RIEMANN-INTEGRAL-001 · ★☆☆ Basic
Compute \(\displaystyle\int_0^1 x\,dx\).
Choose one option.
Show answer
- A. \(\tfrac12\)
Explanation
Lower and upper sums with \(n\) equal parts are \(\tfrac12\mp\tfrac1{2n}\), and both tend to \(\tfrac12\).
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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