Trapping area from both sides

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

For f(x) = x on [0, 1] with partition {0, 0.5, 1}, what is the lower sum?
Answer: ______________
A lower Riemann sum on each subinterval uses what?
- suprema of f on each subinterval
- infima of f on each subinterval
- midpoints of each subinterval
- right endpoints
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 only decrease the upper sum.
Circle one: True False
For f(x) = x squared on [0, 1] with partition {0, 0.5, 1}, 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 partition {0, 0.5, 1}, what is the upper sum?
- 0.75
- 0.25
- 0.5
- 1
The Dirichlet function (1 on rationals, 0 on irrationals) on [0, 1] is not Riemann integrable. True or false?
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 is the Dirichlet function on [0, 1] not Riemann integrable?
- it is unbounded on [0, 1]
- its integral would have to be 0.5
- every interval contains rationals and irrationals, so upper sums stay 1 and lower sums stay 0
- it takes only finitely many values

Answer key

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

0.25 · 0.25. Infima 0 and 0.5 times width 0.5 give 0 + 0.25.
infima of f on each subinterval · Infima of f on each subinterval. Underestimating rectangles stay beneath the graph.
Infima of f on each subinterval · Lower sums take the floor of f on every piece.
True · Refinement lowers the ceiling, never raising it.
0.125 · 0.125. Infima 0 and 0.25 times width 0.5 give 0 + 0.125.
Upper and lower sums can be made arbitrarily close · Integrability means the trap can be squeezed shut.
0.75 · 0.75. Suprema 0.5 and 1 times width 0.5 give 0.25 + 0.5.
True · True. Every piece has supremum 1 and infimum 0, so sums never meet.
True · Its upper sums freeze at 1 and lower sums at 0, so the trap never closes.
every interval contains rationals and irrationals, so upper sums stay 1 and lower sums stay 0 · Every interval contains rationals and irrationals, so upper sums stay 1 and lower sums stay 0. The gap never closes.