Exercise · ZW-PARTITION-AND-DARBOUX-SUMS-009 · ★★☆ Intermediate
Let \(f(x)=x^3\) on \([-1,1]\) and \(P=\{-1,0,1\}\). Compute \(U(f,P)-L(f,P)\).
Choose one option.
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- B. \(2\)
Explanation
\(L=(-1)\cdot1+0\cdot1=-1\) and \(U=0\cdot1+1\cdot1=1\), so \(U-L=2\). The gap is not zero even though the integral is \(0\); it vanishes only upon refining.
Review the concept: Partition and Darboux sums
A partition \(P\) of \([a,b]\) splits it into subintervals of length \(\Delta x_i\); the lower sum \(L\) and upper sum \(U\) of Darboux use the infimum \(m_i\) and the supremum \(M_i\) of \(f\) on each one.
\[L(f,P)=\sum_{i=1}^{n} m_i\,\Delta x_i,\qquad U(f,P)=\sum_{i=1}^{n} M_i\,\Delta x_i\]
More practice problems: partition and darboux sums
- ZW-PARTITION-AND-DARBOUX-SUMS-001 Let f(x)=x on [0,1] and P=\0,tfrac12,1\. Compute L(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-002 With the same f(x)=x and P=\0,tfrac12,1\ on [0,1], compute U(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-003 For f(x)=x^2 on [0,2] and P=\0,1,2\, compute U(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-004 Let f(x)=x^2 on [0,1] and P the partition into 4 equal parts. Compute L(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-005 Let f(x)=-x on [0,1] and P=\0,tfrac12,1\. Compute L(f,P).
- ZW-PARTITION-AND-DARBOUX-SUMS-006 Let f(x)=3 on [1,4] and P any partition. What is U(f,P)?
- ZW-PARTITION-AND-DARBOUX-SUMS-007 If Q is a refinement of P (that is, P⊆ Q), which chain is correct?
- ZW-PARTITION-AND-DARBOUX-SUMS-008 For any two partitions P_1 and P_2 of [a,b], which inequality always holds?
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