MATH1115 Lecture Notes - Lecture 38: Riemann Sum, Jean Gaston Darboux, Greatest And Least Elements
MATH1115: Riemann Sum and Riemann - Darboux Integration
● Riemann - Darboux Integration:
○ Chopped up an interval into small pieces and in order to figure out to the
heights that are over or under.
○ Useful on theory
○ sup{L (f,P) <- just a number: collect the set of all numbers as we let P vary
as a partition of [a,b]} -> lower integral
○ Inf{ U (f,P): P is a partition of [a,b]} -> upper integral
○ We need a theorem that will allow us to tell if something is integrable or
not
● Riemann Sums:
○ Don’t find the upper or lower points you have to take particularly choices
that lie in the subintervals
○ Arbitrary riemann sum doesn’t need to be the midpoint so long as Ci
values are in the partition.
○ You take the value of the function at the input (arbitrary) times by the width
of the rectangle hence riemann sum/integration
○ Useful for practical
○ Doesn’t give you a definite area with functions with an asymptote
○ Relies of the definition that its bounded hence can find the bounded range
of values that it takes.
○ We need to take the supremum and infimum of the set of values
● REVIEW:
○ Every nonempty subset of R which is bounded above has a supremum
○ There can be at most one least upper bound: If M and N are two least
upper bounds then M< or equal to N and N is < or equal to M hence M=N
○ E.g. If A where to equal the closed interval from 0,1 [0,1] then supA = 1
since you take the greatest element of A.
■ To prove this only relying on the definition:
■ Need to justify that 1 is the upper bound and any other upper bound
is equal to it
■ A = { x: x is > or equal to 0 and x is < or equal to 1} = the set of all
the values x such that x is > or equal to 0 and blah
■ So if X is an element of A (xEA) we have that x is < or equal to 1
■ That is 1, is an upper bound for A
■ Now assume that M is any upper bound for A (M = arbitrary fixed
choice)
■ By definition xEA -> X is < or equal to M
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Document Summary
Math1115: riemann sum and riemann - darboux integration. Chopped up an interval into small pieces and in order to figure out to the heights that are over or under. Sup{l (f,p) lower integral. Inf{ u (f,p): p is a partition of [a,b]} -> upper integral. We need a theorem that will allow us to tell if something is integrable or not. Don"t find the upper or lower points you have to take particularly choices that lie in the subintervals. Arbitrary riemann sum doesn"t need to be the midpoint so long as ci values are in the partition. You take the value of the function at the input (arbitrary) times by the width of the rectangle hence riemann sum/integration. Doesn"t give you a definite area with functions with an asymptote.
