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Flashcards: Riemann Integral, Riemann Sums, & Improper Riemann Integration

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a<b $\displaystyle a< b$ , and $f:[a,b]\rightarrow \mathbb{R}$ $\displaystyle f:[a,b]\rightarrow \mathbb{R}$ be bounded

$\displaystyle a$ , $b\ \epsilon\ \mathbb{Z}$ $\displaystyle b\ \epsilon\ \mathbb{Z}$ , a<b $\displaystyle a< b$ , and $f:[a,b]\rightarrow \mathbb{R}$ $\displaystyle f:[a,b]\rightarrow \mathbb{R}$ be bounded

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a>b $\displaystyle a>b$ , and $f:[a,b]\rightarrow \mathbb{Z}$ $\displaystyle f:[a,b]\rightarrow \mathbb{Z}$

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a<b $\displaystyle a< b$ , and $f:[a,b]\rightarrow \mathbb{Z}$ $\displaystyle f:[a,b]\rightarrow \mathbb{Z}$ be bounded

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a>b $\displaystyle a>b$ , and $f:[a,b]\rightarrow \mathbb{R}$ $\displaystyle f:[a,b]\rightarrow \mathbb{R}$

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Flashcards: Riemann Integral, Riemann Sums, & Improper Riemann Integration

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a>b $\displaystyle a>b$ , and $f:[a,b]\rightarrow \mathbb{Z}$ $\displaystyle f:[a,b]\rightarrow \mathbb{Z}$

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a>b $\displaystyle a>b$ , and $f:[a,b]\rightarrow \mathbb{R}$ $\displaystyle f:[a,b]\rightarrow \mathbb{R}$

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Flashcards: Riemann Integral, Riemann Sums, & Improper Riemann Integration

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

Your answer:

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a>b $\displaystyle a>b$ , and $f:[a,b]\rightarrow \mathbb{Z}$ $\displaystyle f:[a,b]\rightarrow \mathbb{Z}$

Your answer:

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a>b $\displaystyle a>b$ , and $f:[a,b]\rightarrow \mathbb{R}$ $\displaystyle f:[a,b]\rightarrow \mathbb{R}$

Correct answer:

Explanation:

Using the definition for Riemann sums to define the upper and lower integrals of a function answers the question.

According the the Riemann sum where U(f,P) $\displaystyle U(f,P)$ represents the upper integral and L(f,P) $\displaystyle L(f,P)$ the following are defined:

1. The upper integral of $\displaystyle f$ on [a,b] $\displaystyle [a,b]$ is

$(U)\int_a^bf(x)dx:=\text{inf}\begin{Bmatrix} U(f,p)) \end{Bmatrix}$ $\displaystyle (U)\int_a^bf(x)dx:=\text{inf}\begin{Bmatrix} U(f,p)) \end{Bmatrix}$ where $\displaystyle P$ is a partition of [a,b] $\displaystyle [a,b]$ .

2. The lower integral of $\displaystyle f$ on [a,b] $\displaystyle [a,b]$ is

$(L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ $\displaystyle (L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ where $\displaystyle P$ is a partition of [a,b] $\displaystyle [a,b]$ .

3. If 1 and 2 are the same then the integral is said to be

$\int_a^bf(x):=(U)\int_a^bf(x)dx=(L)\int_a^bf(x)dx$ $\displaystyle \int_a^bf(x):=(U)\int_a^bf(x)dx=(L)\int_a^bf(x)dx$

if and only if $\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a<b $\displaystyle a< b$ , and $f:[a,b]\rightarrow \mathbb{R}$ $\displaystyle f:[a,b]\rightarrow \mathbb{R}$ be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if $\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ , a<b $\displaystyle a< b$ , and $f:[a,b]\rightarrow \mathbb{R}$ $\displaystyle f:[a,b]\rightarrow \mathbb{R}$ be bounded.

Flashcards: Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

$\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ and a<b $\displaystyle a< b$ . Over the interval [a,b] $\displaystyle [a,b]$ is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ $\displaystyle P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b $\displaystyle a=x_0< x_1< ...< x_n=b$

Upper Riemann sum

Norm

Refinement of a partition

Partition

Lower Riemann sum

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Flashcards: Riemann Integral, Riemann Sums, & Improper Riemann Integration

a=x_0<x_1<...<x_n=b $\displaystyle a=x_0< x_1< ...< x_n=b$

Your answer:

Upper Riemann sum

Your answer:

Norm

Your answer:

Refinement of a partition

Your answer:

Lower Riemann sum

Correct answer:

Partition

Explanation:

By definition

If $\displaystyle a$ , $b\ \epsilon\ \mathbb{R}$ $\displaystyle b\ \epsilon\ \mathbb{R}$ and a<b $\displaystyle a< b$ .

A partition over the interval [a,b] $\displaystyle [a,b]$ is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ $\displaystyle P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b $\displaystyle a=x_0< x_1< ...< x_n=b$ .

Therefore, the term that describes this statement is partition.

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