Explanation Given data: The curve C consists of a position vector function r ( t ) . The limits of scalar parameter is given as a ≤ t ≤ b . Formula used: Write the expression to find ∫ C r ⋅ d r as follows. ∫ C r ⋅ d r = ∫ a b r ( t ) ⋅ r ′ ( t ) d t (1) Write the expression to find r ′ ( t ) of the object. r ′ ( t ) = d d t [ r ( t ) ] (2) Consider the position vector r ( t ) as follows. r ( t ) = 〈 x ( t ) , y ( t ) , z ( t ) 〉 Calculation of r ′ ( t ) : Substitute 〈 x ( t ) , y ( t ) , z ( t ) 〉 for r ( t ) in equation (2), r ′ ( t ) = d d t [ 〈 x ( t ) , y ( t ) , z ( t ) 〉 ] = 〈 d d t [ x ( t ) ] , d d t [ y ( t ) ] , d d t [ z ( t ) ] 〉 = 〈 x ′ ( t ) , y ′ ( t ) , z ′ ( t ) 〉 Calculation of ∫ C r ⋅ d r : Substitute 〈 x ( t ) , y ( t ) , z ( t ) 〉 for r ( t ) , 〈 x ′ ( t ) , y ′ ( t ) , z ′ ( t ) 〉 for r ′ ( t ) , a for a , and b for b in equation (1), ∫ C r ⋅ d r = ∫ a b 〈 x ( t ) , y ( t ) , z ( t ) 〉 ⋅ 〈 x ′ ( t ) , y ′ ( t ) , z ′ ( t ) 〉 d t = ∫ a b [ x ( t ) x ′ ( t ) + y ( t ) y ′ ( t ) + z ( t ) z ′ ( t ) ] d t = ∫ a b

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ISBN: 9781337613927

Author: Stewart

Publisher: CENGAGE L

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Chapter 16.2, Problem 52E

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To show: The expression ∫Cr⋅dr=12[|r(b)|2−|r(a)|2] .

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Chapter 16.2, Problem 51E

Chapter 16.2, Problem 53E

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Chapter 16 Solutions

Calculus, Early Transcendentals

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The expression ∫ C r ⋅ d r = 1 2 [ | r ( b ) | 2 − | r ( a ) | 2 ] .

Solution Summary: The expression C consists of a position vector function r(t).

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To show: The expression ∫Cr⋅dr=12[|r(b)|2−|r(a)|2] .

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Chapter 16 Solutions