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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Integration How do you find the integral of a vector-valued function?

To determine

The value of the integral of a vector-valued function.

Explanation

Let r(t)=f(t)i+g(t)j+h(t)k be the vector-valued function.

If f(t), g(t) and h(t) are continuous on the closed interval [a,b] where atb, then the indefinite integral of the function r(t) is calculated component-by-component as,

r(t)dt=[f(t)dt]i+[g(t)dt]j+[h(t)dt]k=[F(t)+C1]i+[G(t)+C2]j+[H(t)+C3]k=F(t)i+G(t)j+H(t)k+C1i+C2j+C3k=F(t)i+G(t)j+H(t)k+C

Where F(t)=f(t)dt, G(t)=g(t)dt and H(t)=h(t)dt and C=C1i+C2j+C3k is an arbitrary constant vector.

For example,

Consider the indefinite integral:

r(t)dt=(3t2i+j2tk)dt

Integrate the above indefinite integral component-by-component to get:

(3t2i+j2tk)dt=[3t2dt]i+[dt]j+[(2t)dt]k=3(t33)i+tj2(t22)k+(C1i+C2j+C3k)=t3i+tjt2k+C

Where,C is the constant of integration

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Chapter 12 Solutions

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P-93ESect-12.5 P-94ESect-12.5 P-95ESect-12.5 P-96ECh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-39RECh-12 P-40RECh-12 P-41RECh-12 P-42RECh-12 P-43RECh-12 P-44RECh-12 P-45RECh-12 P-46RECh-12 P-47RECh-12 P-48RECh-12 P-49RECh-12 P-50RECh-12 P-51RECh-12 P-52RECh-12 P-53RECh-12 P-54RECh-12 P-55RECh-12 P-56RECh-12 P-57RECh-12 P-58RECh-12 P-59RECh-12 P-60RECh-12 P-61RECh-12 P-62RECh-12 P-63RECh-12 P-64RECh-12 P-65RECh-12 P-66RECh-12 P-67RECh-12 P-68RECh-12 P-69RECh-12 P-70RECh-12 P-71RECh-12 P-1PSCh-12 P-2PSCh-12 P-3PSCh-12 P-4PSCh-12 P-5PSCh-12 P-6PSCh-12 P-7PSCh-12 P-8PSCh-12 P-9PSCh-12 P-10PSCh-12 P-11PSCh-12 P-12PSCh-12 P-13PSCh-12 P-14PS

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