11) Integration Flashcards
What is the relationship between a function bounded on two intervals
Desccribe the proof of a function bound on two intervals
What is the Upper Integral
What is the Lower Integral
What is a refinement of a partiton
Describe the proof that L(P, f) ≤ L(D, f)
and U(D, f) ≤ U(P, f)
Describe the proof that L(Q, f) ≤ U(R, f).
Describe the proof that
When is a function Riemann integrable
Describe the proof that
What are some properties that imply a function is Riemann integrable
- Monotnic function
- Continuous function
- If for all ε > 0 there exists a partition Pε such that U(Pε, f) − L(Pε, f) < ε
Describe the proof that a function is
Reimann integrable when monotonic
Describe the proofthat a function is
Reimann integrable when continuous
If f(x) ≤ g(x) for all x ∈ [a, b] then what can
we say about their integrals
What can we say about the modulus on the a function’s integral
Describe the proof that the modulus of an integral is less than or equal to the integral of the modulus of a function
If f is Riemann integrable is f^2 Riemann integrable
Yes - If f is bounded and Riemann integrable on [a, b] then f2 is Riemann integrable on [a, b].
Is the product of 2 Riemann integrable functions Riemann integrable
Yes - If f, g are bounded and Riemann integrable on [a, b] then the product fg is Riemann integrable on [a, b].
What is the theorem of subintervals of Riemann integrable bounds
If f is bounded and Riemann integrable on [a, b] then f is
Riemann integrable over any sub-interval [c, d] ⊆ [a, b]. Also, for any c ∈ [a, b] we have
