To determine Evaluate the Line integral ∫ C ( x + y ) d s over the paths C 1 : y = x 2 from ( 0 , 0 ) to ( 1 , 1 ) and C 2 : y = x from ( 1 , 1 ) to ( 0 , 0 ) . Explanation Given information: The Line integral ∫ C ( x + y ) d s over the paths C 1 : y = x 2 from ( 0 , 0 ) to ( 1 , 1 ) and C 2 : y = x from ( 1 , 1 ) to ( 0 , 0 ) . Calculation: Write the Line integral. ∫ C ( x + y ) d s (1) Find the smooth parameterization of C 1 : y = x 2 . C 1 : r ( t ) = t i + t 2 j , 0 ≤ t ≤ 1 (2) The components have continuous first derivatives. Differentiate Equation (2) with respect to t . d r d t = ( 1 ) i + ( 2 t ) j = i + 2 t j (3) Find the value of | d r d t | . | d r d t | = ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 2 = ( 1 ) 2 + ( 2 t ) 2 = 1 + 4 t 2 (4) Find the function f ( x , y ) = f ( x + y ) . Substitute t for x and t 2 for y in f ( x , y ) = f ( x + y ) . f ( x , y ) = f ( x + y ) = t + t 2 = t + t = 2 t (5) Evaluate the integral ∫ C 1 f ( x , y ) d s = ∫ C 1 ( x + y ) d t . Substitute Equation (4) and Equation (5) in ∫ C 1 f ( x , y ) d s = ∫ C 1 ( x + y ) d t . ∫ C 1 f ( x , y ) d s = ∫ C 1 ( x + y ) d t = ∫ 0 1 ( 2 t ) ( 1 + 4 t 2 ) d t = 2 ∫ 0 1 t ( 1 + 4 t 2 ) 1 2 d t = [ 1 6 ( 1 + 4 t 2 ) 3 2 ] 0 1 ∫ C 1 f ( x , y ) d s = 1 6 [ ( 1 + 4 ( 1 ) 2 ) 3 2 − ( 1 + 4 ( 0 ) 2 ) 3 2 ] = 1 6 [ ( 1 + 4 ) 3 2 − ( 1 ) 3 2 ] = 1 6 [ ( 5 ) 3 2 − 1 ] = 5 5 − 1 6 Find the smooth parameterization of C 2 : y = x . C 2 : r ( t ) = ( 1 − t ) i + ( 1 − t ) j , 0 ≤ t ≤ 1 (6) The components have continuous first derivatives. Differentiate Equation (6) with respect to t . d r d t = ( 0 − 1 ) i + ( 0 − 1 ) j = − i − j (7) Find the value of | d r d t |

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ISBN: 9781323835029

Author: WEIR

Publisher: PEARSON C

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Chapter 16.1, Problem 25E

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Evaluate the Line integral ∫C(x+y)ds over the paths C1:y=x2 from (0,0) to (1,1) and C2:y=x from (1,1) to (0,0).

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Chapter 16.1, Problem 24E

Chapter 16.1, Problem 26E

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