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Trapping area from both sides

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

For f(x) = x on [0, 1] with four equal subintervals, what is the lower sum?
Answer: ______________
For any bounded function and partition, which inequality must hold?
- The lower sum is at most the upper sum
- The lower sum equals the upper sum after one refinement
- The upper sum is at most the lower sum of any refinement
- The lower sum is at least the left-endpoint Riemann sum
A lower Riemann sum on each subinterval uses what?
- Suprema of f on each subinterval
- Infima of f on each subinterval
- Right endpoints
Refining a partition can never increase the upper sum.
Circle one: True False
For f(x) = x squared on [0, 1] with four equal subintervals, what is the lower sum?
Answer: ______________
The Riemann integrability criterion says what?
- Every partition gives equal sums
- Upper sums increase with refinement
- Upper and lower sums can be made arbitrarily close
For f(x) = x on [0, 1] with four equal subintervals, what is the upper sum?
- 0.625
- 0.375
- 0.5
- 1
For the Dirichlet function on [0, 3], the upper-lower gap is 3 for every partition. Is this true?
Circle one: True False
The Dirichlet function (1 on rationals, 0 on irrationals) on [0, 1] is not Riemann integrable.
Circle one: True False
Why does continuity on a closed interval imply Riemann integrability?
- Fine partitions make the upper-lower gap arbitrarily small
- Continuity makes every Riemann sum equal the exact integral
- Continuity makes the lower sum equal the upper sum for every partition
- A continuous function is monotone, so endpoints give the sums

Answer key

For grown-ups. Fold this page away before handing over the rest.

Trapping area from both sides W1-mt_Ahl2MYIA4N-s1

0.375 · The lower heights are 0, 0.25, 0.5, and 0.75. Multiply their sum by width 0.25 to get 0.375.
The lower sum is at most the upper sum · Each infimum is at most the corresponding supremum, so the lower sum cannot exceed the upper sum.
Infima of f on each subinterval · Lower sums use the infimum of f, its greatest lower bound, on every piece.
True · Refinement lowers the upper sum or leaves it unchanged, but never raises it.
0.21875 · The lower heights are 0, 0.0625, 0.25, and 0.5625. Their sum times width 0.25 is 0.21875.
Upper and lower sums can be made arbitrarily close · Integrability means the trap can be squeezed shut.
0.625 · The upper heights are 0.25, 0.5, 0.75, and 1. Their sum times width 0.25 is 0.625.
True · Every piece has supremum 1 and infimum 0, so it adds its width to the gap. The widths add to 3, so the upper sum is 3 and the lower sum is 0.
True · Its upper sums freeze at 1 and lower sums at 0, so the trap never closes.
Fine partitions make the upper-lower gap arbitrarily small · Uniform continuity on a closed interval makes the oscillation on each sufficiently small piece small, so the upper-lower gap can be squeezed below any positive number.