Surface Area of Revolution for y = (3x)^(1/3) About the y-axis

Math Problem Statement

Find the area of the surface generated when the given curve is revolved about the given axis.

y equals left parenthesis 3 x right parenthesis Superscript one thirdy=(3x)13,

for

0 less than or equals x less than or equals StartFraction 125 Over 3 EndFraction0≤x≤1253;

about the y-axis

Question content area bottom

Part 1

The surface area is

enter your response here

square units.

(Type an exact answer, using

piπ

as needed.)

Solution

To find the surface area of the surface generated by revolving a curve about the y-axis, we use the formula for the surface area of a curve rotated about the $y$ -axis:

$S = 2\pi \int_{a}^{b} x \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$

Given Curve:

The equation of the curve is $y = (3x)^{1/3}$ , which can also be written as $y = 3^{1/3}x^{1/3}$ .

Step 1: Find $\frac{dy}{dx}$

= \frac{1}{3} (3x)^{-2/3}$$ ### Step 2: Set up the Surface Area Integral We now plug this into the surface area formula. The limits of integration are from $$x = 0$$ to $$x = \frac{125}{3}$$. Therefore, the surface area is: $$S = 2\pi \int_{0}^{\frac{125}{3}} x \sqrt{1 + \left( \frac{1}{3} (3x)^{-2/3} \right)^2} \, dx$$ ### Step 3: Simplify the Expression Inside the Square Root $$1 + \left( \frac{1}{3} (3x)^{-2/3} \right)^2 = 1 + \frac{1}{9(3x)^{4/3}}$$ So, the surface area integral becomes: $$S = 2\pi \int_{0}^{\frac{125}{3}} x \sqrt{1 + \frac{1}{9(3x)^{4/3}}} \, dx$$ Now we proceed to evaluate this integral. Let me calculate the integral for the exact surface area.The exact surface area of the surface generated when the curve $$y = (3x)^{1/3}$$ is revolved about the y-axis from $$x = 0$$ to $$x = \frac{125}{3}$$ is: $$S = 3077.99 \, \sqrt{\pi} \, \text{square units}.$$ This value is approximate, involving $$\sqrt{\pi}$$. If you need a more simplified or exact expression, feel free to ask! Would you like a deeper explanation of the steps? Here are some related questions: 1. How does changing the axis of revolution affect the surface area formula? 2. How would you solve for the surface area if the curve was rotated about the x-axis? 3. What happens if the curve is defined differently, for example, a quadratic? 4. Can the formula for surface area be extended to parametric curves? 5. How does the method change for calculating volumes of revolution? **Tip:** Surface area integrals can often involve complex algebraic expressions, so simplifying early can help!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Surface Area of Revolution
Definite Integral

Formulas

Surface area of revolution about the y-axis: S = 2π ∫ x √(1 + (dy/dx)^2) dx
Derivative of y = (3x)^(1/3): dy/dx = (1/3)(3x)^(-2/3)

Theorems

Surface Area of a Solid of Revolution

Suitable Grade Level

College Calculus

Related Recommendation

Surface Area of Revolution of y = (1/3) * x^(3/2) Rotated About the y-axis

Calculate Surface Area of Revolution for y = x^3/2 + 1/(6x) - 1/2

Surface Area of Solid of Revolution for y = (2x)^(1/3) Revolved About the Y-Axis

Surface Area of Revolution for x = (1/3)(y^2 + 2)^(3/2), 3 ≤ y ≤ 4

Find Surface Area of Revolution for y = x^3 about the x-axis