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Find the area of the surface obtained by rotating the curve y³=x about the y-axis for 1≤y≤4.

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Final Answer:

The area of the surface formed by rotating the curve y³=x about the y-axis for 1≤y≤4 is A = 48π square units.

Step-by-step explanation:

Given curve: y³ = x

To find the surface area formed by rotating this curve about the y-axis within the interval 1 ≤ y ≤ 4, use the formula for surface area of revolution:


\[ A = 2π \int_(1)^(4) f(y) √(1 + (f'(y))^2) dy \]

Given the curve y³ = x, express x in terms of y:
x = y^(2/3).

Find the derivative of x with respect to
y: \( (dx)/(dy) = (2)/(3) y^(-1/3) \).

Substitute these values into the surface area formula:


\[ A = 2π \int_(1)^(4) y^(2/3) \sqrt{1 + \left((2)/(3) y^(-1/3)\right)^2} dy \]

Simplify the integrand and solve for the surface area. Introduce a substitution where
\( u = 1 + (4)/(9)y^(2/3) \), which transforms the integral to:


\[ A = 2π \int_(13/9)^(25/9) u^(1/2) du \]

Integrate with respect to u:


\[ A = 2π \left[(2)/(3) u^(3/2)\right]_(13/9)^(25/9) \]

After evaluating the integral, the final result is:


\[ A = 48π \]

Therefore, the surface area obtained by rotating the curve y³ = x about the y-axis for 1 ≤ y ≤ 4 is 48π square units.

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