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Q. 11

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Calculus
Found in: Page 871
Calculus

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

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

A vector function is integrable on the interval [a,b] if each of its componentfunctions 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 forthe 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).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

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 iflimn→∞∑k=1∞f(xkn) ∆x exists where ∆x=b-an, xk=a+k∆x and xkn =xk-1,xk

Step 3. Result

A vector function is integrable on the interval [a,b] if each of its componentfunctions 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 forthe 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).

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