11) Integration

Question 1

What is the relationship between a function bounded on two intervals

Answer 1

Question 2

Desccribe the proof of a function bound on two intervals

Answer 2

Question 3

What is the Upper Integral

Answer 3

Question 4

What is the Lower Integral

Answer 4

Question 5

What is a refinement of a partiton

Answer 5

Question 6

Describe the proof that L(P, f) ≤ L(D, f)
and U(D, f) ≤ U(P, f)

Answer 6

Question 7

Describe the proof that L(Q, f) ≤ U(R, f).

Answer 7

Question 8

Describe the proof that

Answer 8

Question 9

When is a function Riemann integrable

Answer 9

Question 10

Describe the proof that

Answer 10

Question 11

What are some properties that imply a function is Riemann integrable

Answer 11

• Monotnic function
• Continuous function
• If for all ε > 0 there exists a partition Pε such that U(Pε, f) − L(Pε, f) < ε

Question 12

Describe the proof that a function is
Reimann integrable when monotonic

Answer 12

Question 13

Describe the proofthat a function is
Reimann integrable when continuous

Answer 13

Question 14

If f(x) ≤ g(x) for all x ∈ [a, b] then what can
we say about their integrals

Answer 14

Question 15

What can we say about the modulus on the a function’s integral

Answer 15

Question 16

Describe the proof that the modulus of an integral is less than or equal to the integral of the modulus of a function

Answer 16

Question 17

If f is Riemann integrable is f^2 Riemann integrable

Answer 17

Yes - If f is bounded and Riemann integrable on [a, b] then f2 is Riemann integrable on [a, b].

Question 18

Is the product of 2 Riemann integrable functions Riemann integrable

Answer 18

Yes - If f, g are bounded and Riemann integrable on [a, b] then the product fg is Riemann integrable on [a, b].

Question 19

What is the theorem of subintervals of Riemann integrable bounds

Answer 19

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