/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Find the area of the surface gen... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 17

Find the area of the surface generated when the given curve is revolved about the \(y\) -axis. $$y=(3 x)^{1 / 3}, \text { for } 0 \leq x \leq \frac{8}{3}$$

Short Answer

Expert verified
Answer: The area of the surface generated is \(2\pi\left[6 - \ln(8)\right]\).

Step by step solution

01

Find the derivative of f(x)

To find the derivative of \(f(x) = (3x)^{1/3}\), we will apply the power rule for differentiation, which gives us: $$f'(x)=\frac{d}{dx}(3x)^{1/3} = \frac{1}{3}(3x)^{-2/3}(3)$$
02

Plug the derivative into the formula

Now we plug our original function \(f(x)\) and its derivative \(f'(x)\) into the surface area formula: $$S = 2 \pi \int_0^{\frac{8}{3}} (3x)^{1/3} \sqrt{1 + [(\frac{1}{3}(3x)^{-2/3}(3))^2]} dx$$
03

Simplify the integrand

We now simplify the integrand by squaring the derivative and adding 1: $$1 + [(\frac{1}{3}(3x)^{-2/3}(3))^2] = 1 + (\frac{9}{9x^{4/3}})=\frac{9x^{4/3} + 9}{9x^{4/3}} = [1+(3x)^{-4/3}]$$ Thus, the integral becomes: $$S = 2 \pi\int_0^{\frac{8}{3}}(3x)^{1/3} [1+(3x)^{-4/3}] dx$$
04

Evaluate the integral

Now, we evaluate the integral by breaking it into two parts: $$S = 2 \pi \left[\int_0^{\frac{8}{3}}(3x)^{1/3} dx + \int_0^{\frac{8}{3}}(3x)^{1 / 3}(3x)^{-4 / 3} dx\right]$$ For the first integral, we can use a simple substitution \(u=3x\) and proceed with the integration. For the second integral, notice how the powers combine to make a simpler function: $$(3x)^{1 / 3}(3x)^{-4 / 3} = (3x)^{1/3-4/3} = (3x)^{-1} = \frac{1}{3x}$$ Now we can evaluate both integrals: $$S = 2\pi\left[\int_0^{\frac{8}{3}}(3x)^{1/3} dx + \int_0^{\frac{8}{3}}\frac{1}{3x} dx\right] = 2\pi\left[\frac{3}{4}(3x)^{4/3}\Bigg|_0^{\frac{8}{3}} - \ln(3x)\Bigg|_0^{\frac{8}{3}}\right]$$
05

Calculate the surface area

Now we plug in our limits of integration and calculate the surface area: $$S = 2\pi\left[\frac{3}{4}(3\left(\frac{8}{3}\right))^{4/3} - \ln (3(\frac{8}{3})) - \frac{3}{4}(3(0))^{4/3} + \ln(3(0))\right] = 2\pi\left[\frac{3}{4}(8) - \ln(8)\right]$$ $$S = 2\pi\left[6 - \ln(8)\right]$$ So the area of the surface generated when the given curve is revolved about the \(y\)-axis is equal to \(2\pi\left[6 - \ln(8)\right]\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the mass of the following thin bars with the given density function. $$\rho(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } 0 \leq x \leq 1 \\ x(2-x) & \text { if } 1

It takes \(100 \mathrm{J}\) of work to stretch a spring \(0.5 \mathrm{m}\) from its equilibrium position. How much work is needed to stretch it an additional \(0.75 \mathrm{m} ?\)

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right)$$

Evaluate the following integrals. $$\int_{0}^{1} \frac{16^{x}}{4^{2 x}} d x$$

Alternative proof of product property Assume that \(y>0\) is fixed and that \(x>0 .\) Show that \(\frac{d}{d x}(\ln x y)=\frac{d}{d x}(\ln x) .\) Recall that if two functions have the same derivative, they differ by an additive constant. Set \(x=1\) to evaluate the constant and prove that \(\ln x y=\ln x+\ln y.\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.