A nonintegrable function Consider the function defined on [0, 1] such that f ( x ) − 1 if x is a rational number and f ( x ) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral ∫ 0 1 f ( x ) d x does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n . ( Hint: Between any two real numbers lie a rational and an irrational number.)
A nonintegrable function Consider the function defined on [0, 1] such that f ( x ) − 1 if x is a rational number and f ( x ) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral ∫ 0 1 f ( x ) d x does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n . ( Hint: Between any two real numbers lie a rational and an irrational number.)
Solution Summary: The author shows the right, left and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all.
A nonintegrable function Consider the function defined on [0, 1] such that f(x) − 1 if x is a rational number and f(x) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral
∫
0
1
f
(
x
)
d
x
does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n. (Hint: Between any two real numbers lie a rational and an irrational number.)
1) It shows that the function f(x) = 1/x is integrable in the closed interval [1, 2]. This must be done estimating the Riemann sums of the function f(x).
2)Propose a limited function f : [a, b] → R and two P and Q partitions of the closed interval [a, b] suchthat λ(P) < λ(Q) but that lower sum (f; P) = lower sum(f; Q) and superior sum (f; P) = superior sum(f; Q).
3)Let f : [a, b] → R a continuous function such that f(x) ≥ 0 for all x ∈ [a, b]. It proves that if there isx0 ∈ (a, b) such that f(x0)>0, then:
The integral of a to b of f(x) is greater than zero.
Note: do not demonstrate 3) with a numerical example
4)Give an example of a dimensioned function f : [a, b] → R such that |f| is Riemann-integratable, but f notis Riemann-integrable.
5)Build two examples of functions f1,2 : [a, b] → R that are Riemann-integratable where1/f1(X)is not Riemann-integrable and 1/f2(X)if Riemann-integrable
Consider the function f(x)=x2on the interval [1,9]. Let P be a uniform partition of [1,9] with 16 sub-intervals. Compute the left and right Riemann sum of f on the partition. Use exact values.
Left-sum: Right-sum:
Find a formula for the riemann sum obtained by dividing the interval [a,b] into n equal subintervals and using ci=a+ithetax. then take the limit of the sums as n to infinity.
show all work
f(x)=x^2+1 over the interval [0,3]
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