Pure Mathematics F3: Integration

17 flashcards · Mathematics · by lakesare

  1. Visually imagine: discontinuous function that can be integrated discontinuous function that can NOT be integrated.

    can be integrated can NOT be integrated Dirichlet function We can always assign a value to the area between the graph and the x-axis for a continuous function defined on a closed interval. You will also see that we can do the same for the function in (a), but not for the function in (b).

  2. Supremum of a set - sup(E) VS supremum of a function - sup(f)?

    Sup(f) is a supremum of the image set of f. The concepts of the supremum of a real function and the supremum of a set of real numbers are related in the following way: the supremum of a function is the supremum of the image set of the function.

  3. How does this notation expand?

  4. What's inf(f)?

    inf(f) = –1

  5. [Partition of an interval] Visually imagine the partition P of [a, b].

  6. [Partition of an interval] Visually imagine δxi.

    δxi is the length of xi–1... xi subinterval.

  7. [Partition of an interval] What's ║P║ called?

    Mesh of P.

  8. [Partition of an interval] What's a mesh of P?

    The length of the largest subinterval.

  9. [Partition of an interval] Partition VS standard partition of an interval?

    partition - different sizes standard partition - equal sizes

  10. [Riemann Integral] Visually imagine the Lower Riemann Sum.

  11. [Riemann Integral] What's Mi and δxi in the upper Riemann sum definition?

    Mi - supremum of f on this interval δxi - length of this interval

  12. [Riemann Integral] What do we call these areas?

    Lower Riemann sum - L(f, P) Upper Riemann sum - U(f, P)

  13. [Riemann Integral] What's shocking about Lower and Upper Riemann sums?

    No matter what the partitions are - L(f, P) is always smaller than U(f, P')!

  14. [Riemann Integral] What do we call a and b?

    Limits of integration.

  15. [Riemann Integral] Lower Riemann Sum VS Lower Riemann Integral

    Lower Riemann Sum - colored area Lower Riemann Integral - the largest area among all partitions

  16. [Riemann Integral] What's P in the definition of a lower integral?

    All possible partitions!

  17. [Integrals] Name 2 famous ways to define integration.

    Riemann integration - vertical rectangles Lebesgue integration - horizontal rectangles