How do you find the length of the curve y = ( 2 x + 1 ) <mrow class="MJX-T

DIAMMIBENVERMk1 2022-07-04 Answered
How do you find the length of the curve y = ( 2 x + 1 ) 3 2 , 0 x 2?
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Answers (2)

Ordettyreomqu
Answered 2022-07-05 Author has 22 answers
The length of a curve between a and b values for x is given by
L = a b 1 + y 2 d x
y 2 = 18 x + 9
L = 0 2 18 x + 10 d x = 1 27 ( 18 x + 10 ) 3 2 ] 0 2 = 10.38
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pablos28spainzd
Answered 2022-07-06 Author has 4 answers
The Arc Length of a curve y=f(x) from x=a to x=b is given by:
L = a b 1 + ( d y d x ) 2 d x
So, for the given curve y = ( 2 x + 1 ) 3 2 for x [ 0 , 2 ] we form the derivative using the power rule for differentiation in conjunction with the chain rule:
d y d x = ( 3 2 ) ( 2 x + 1 ) 3 2 1 ( 2 )
= 3 ( 2 x + 1 ) 1 2
So then, the arc length is:
L = 0 2 1 + ( 3 ( 2 x + 1 ) 1 2 ) 2 d x
= 0 2 1 + 9 ( 2 x + 1 ) d x
= 0 2 1 + 18 x + 9 d x
= 0 2 18 x + 10 d x
And using the power rule for integration, we can integrate to get:
L = [ ( 18 x + 10 ) 3 2 3 2 1 18 ] 0 2
= [ 1 27 ( 18 x + 10 ) 3 2 ] 0 2
= 1 27 [ ( 36 + 10 ) 3 2 10 3 2 ]
= 1 27 [ 46 46 10 10 ]
10.384 ( 3 d p )
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