Find the exact length of the curve x = 1/3 √y (y − 3), 16 ≤ y ≤ 25.

We are going to use using the formula of the exact length of the curve to solve this.

Answer: The exact length of the curve x = 1/3 √y (y − 3), 16 ≤ y ≤ 25 is 64/3.

Let's solve this step by step.

Explanation:

Given, x = 1/3 √y (y − 3), 16 ≤ y ≤ 25

Length of the curve y = f(x) from x =a to x = b is given by: \(\int\limits_a^b\) √1 + [f′(x)]² dx.

x = 1/3 √y (y − 3)

x = 1/3 . y^3/2- y^1/2

Let's find the first derivative of x.

dx/dy = 1/3. 3/2. y^1/2- 1/2 . y^1/2

dx/ dy = 1/2 ( y^1/2-y^-1/2)

(dx/dy)² = 1/4 ( y^1/2-y^-1/2)²

= 1/4(y + y^-1-2)

1 + {f′(x)}²= 1 + 1/4(y + y^-1-2)

= 1/4 y + 1/4y + 1/2

1 + {f′(x)}²= 1/4 (y + y^-1 + 2)

√[1 + {f′(x)}²] = 1/2 ( y^1/2+ y^-1/2)

Length of curve = \(\int\limits_{16}^{25}\) √[1 + {f′(x)}²] dx

= \(\int\limits_{16}^{25}\) √y/2 + 1/2√y dy

= [(y)^3/2/3 + √y]\(^{25}_{16}\)

= [(25)^3/2/3 + √25] - [(16)^3/2/3 + √16]

= [125/3 + 5] - [64/3 + 4]

= 140/3 - 76/3

= (140 - 76)/3

= 64/3