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:
![\[ A = 2π \int_(1)^(4) f(y) √(1 + (f'(y))^2) dy \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/xth7wj6uz9exa2lb9044q0hrpye7mjyb8a.png)
Given the curve y³ = x, express x in terms of y:
.
Find the derivative of x with respect to
.
Substitute these values into the surface area formula:
![\[ A = 2π \int_(1)^(4) y^(2/3) \sqrt{1 + \left((2)/(3) y^(-1/3)\right)^2} dy \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/oih9y4ya1wt8z50sp06bx0gszwwlqqxa3g.png)
Simplify the integrand and solve for the surface area. Introduce a substitution where
, which transforms the integral to:
![\[ A = 2π \int_(13/9)^(25/9) u^(1/2) du \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/a08zusdn0xkfl739wbip5iis35324c6qfj.png)
Integrate with respect to u:
![\[ A = 2π \left[(2)/(3) u^(3/2)\right]_(13/9)^(25/9) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/46qn01i4f7y780qrkdpy1ihi2ce6vvaz6p.png)
After evaluating the integral, the final result is:
![\[ A = 48π \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/9bzy2kltwrx58x7xtisl9lmu20n0zxigvw.png)
Therefore, the surface area obtained by rotating the curve y³ = x about the y-axis for 1 ≤ y ≤ 4 is 48π square units.