Chapter 11: Q. 9 (page 871)

State what it means for a scalar function $y = f (x)$ to be integrable on an interval $[a, b]$

Short Answer

Expert verified

The function is integrable if $\lim_{n \to \infty} \sum_{k = 1}^{n} f (x_{k}^{n}) ∆ x$ exists where $∆ x = \frac{b - a}{n}, x_{k} = a + k ∆ x$ and $x_{k}^{n} = [x_{k - 1}, x_{k}]$

Step by step solution

Step 1. Given information

The given scalar function $y = f (x)$

Step 2. The objective is to define the integrability of y=f(x) on the interval a,b

Definition of Integrability of

y = f (x)

[a, b]

:
Let

f : [a, b] \to R

. Let

f

has on an anti derivative F on $I$ .
Then we say that f has an integral on I and for any real constant c, we say

F + c

an indefinite integral of f over

[a, b]

denoted by

\int f (x) d x

. Thus

\int f = \int f (x) d x = F (x) + c

Where ' c' is called the constant of integration, (or) we can define in another way.
Definition:
Let

f (x)

be a function defined on the interval

[a, b]

. The function is integrable ifrole="math" localid="1649618811563"

\lim_{n \to \infty} \sum_{k = 1}^{n} f (x_{k}^{n}) ∆ x

exists where
role="math" localid="1649618839736"

∆ x = \frac{b - a}{n}, x_{k} = a + k ∆ x

androle="math" localid="1649618859556"

x_{k}^{n} = [x_{k - 1}, x_{k}]

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State what it means for a scalar function $y = f (x)$ to be integrable on an interval $[a, b]$

Short Answer

Step by step solution

Step 1. Given information

Step 2. The objective is to define the integrability of y=f(x) on the interval a,b

One App. One Place for Learning.

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State what it means for a scalar function y=f(x)to be integrable on an interval a,b

Short Answer

Step by step solution

Step 1. Given information

Step 2. The objective is to define the integrability of y=f(x) on the interval a,b

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Probability and Statistics

Statistics

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

State what it means for a scalar function $y = f (x)$ to be integrable on an interval $[a, b]$