Chapter 11: Q. 11 (page 871)

$Explain why we do not need to explicitly use a limit in the definition for the definite integral of a vector function r (t) = <x (t), y (t), z (t)> .$

Short Answer

Expert verified

$A vector function is integrable on the interval [a, b] if each of its component functions is integrable on [a, b]. Every component function r (t) is a scalar function which is defined . The limit is already involved in the definition for the integrability of a scalar function [a, b] . Hence, we need not explicitly use a limit in the definition for the definite integral of r (t) .$

Step by step solution

Step 1. Given information is

$r (t) = <x (t), y (t), z (t)>$

Step 2. Definition of integrabilti on the scalar function

$Integrability of y = f (x) on [a, b] : Let y = f (x) be a function defined on the interval [a, b] . This function is integrable if \lim_{n \to \infty} \sum_{k = 1}^{\infty} f (x_{k}^{n}) ∆ x exists where ∆ x = \frac{b - a}{n}, x_{k} = a + k ∆ x and x_{k}^{n} = [x_{k - 1}, x_{k}]$

Step 3. Result

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$Explain why we do not need to explicitly use a limit in the definition for the definite integral of a vector function r (t) = <x (t), y (t), z (t)> .$

Short Answer

Step by step solution

Step 1. Given information is

Step 2. Definition of integrabilti on the scalar function

Step 3. Result

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Explainwhywedonotneedtoexplicitlyusealimitinthedefinitionforthedefiniteintegralofavectorfunctionr(t)=x(t),y(t),z(t).

Short Answer

Step by step solution

Step 1. Given information is

Step 2. Definition of integrabilti on the scalar function

Step 3. Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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$Explain why we do not need to explicitly use a limit in the definition for the definite integral of a vector function r (t) = <x (t), y (t), z (t)> .$