Exercise · ZW-RIEMANN-INTEGRAL-002 · ★☆☆ Basic
Compute \(\displaystyle\int_0^2 x^2\,dx\).
Choose one option.
Show answer
- D. \(\tfrac83\)
Explanation
It is the common value of the supremum of lower sums and the infimum of upper sums: \(\tfrac{2^3}{3}=\tfrac83\).
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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