The given curve y = \sqrt[3] x, 1 \leq x \leq 27 is rotated about the y-axis. Find the area of...

Question:

The given curve {eq}y = \sqrt[3] x, 1 \leq x \leq 27 {/eq} is rotated about the y-axis. Find the area of the resulting surface.

The Area of a Surface of Revolution:

The area {eq}S {/eq} of the surface obtained by rotating the function {eq}f(x) {/eq} about the x-axis, {eq}x \in [a, b] {/eq} is calculated by using the following formula

{eq}S = 2\pi \int_{a}^{b}f(x)\sqrt{1+(f'(x))^2} \ dx {/eq}

Answer and Explanation:

Step 1: Find the derivative of the function

{eq}f(x) = x^{\frac{1}{3}}\\ f'(x) = \frac{1}{3}x^{-\frac{2}{3}} {/eq}

Step 2: Use the following...

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Learn more about this topic:

How to Find Volumes of Revolution With Integration

from Math 104: Calculus

Chapter 14 / Lesson 5
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