

Find the exact length of the curve. x = 1/3√y(y - 3), 4 ≤ y ≤ 16.
Solution:
Given, x = 1/3 √y (y - 3), 4 ≤ y ≤ 16
Length of the curve, x = f(y) from y =a to y = b is given by:
∫ba√1+f′(y)2.dy
x = 1/3 √y (y - 3)
x = 1/3 (y3/2- 3y1/2)
dx/dy = 1/3 [(3/2)y1/2- (3/2)y-1/2]
= 1/3 [(3y1/2/2) - (3/2y1/2)]
= 1/2 [y1/2 - 1/y1/2]
= (y-1)/2√y = f'(y)
Length of the curve
=∫ba√1+f′(y)2.dy
=∫164√1+(y−12√y)2.dy
=∫164√1+y2−2y+14y.dy
=∫164√4y+y2−2y+14y.dy
=∫164(y+1)2(2√y)2
=∫164y+12√y.dy
=∫16412√y.dy+∫16412√y.dy
=∫16412y12.dy+∫16412y−12.dy
=12y3/23/2|164+12y1/21/2|164
= [1/3 (64-8)] +[4-2]
= (56/3)+2
= 62/3
Find the exact length of the curve. x = 1/3√y(y − 3), 4 ≤ y ≤ 16.
Summary:
The exact length of the curve x = 1/3 √y (y - 3), 4 ≤ y ≤ 16, is 62/3.
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