Q. 65 In Exercises 63鈥�72, set up and... [FREE SOLUTION]

Chapter 6: Q. 65 (page 540)

In Exercises 63鈥�72, set up and solve a definite integral to find the exact area of each surface of revolution obtained by revolving the curve y = f(x) around the x-axis on the interval [a, b].
$f (x) = \sqrt{x}, [a, b] = [0, 4]$

Short Answer

Expert verified

The exact area of the surface of the revolution obtained by revolving the curve $f (x) = \sqrt{x}$ around thex-axis on the interval $[0, 4]$ is $S = \frac{蟺}{6} (17 \sqrt{17} - 1) .$

Step by step solution

Step 1. Given Information.

The given curve is $f (x) = \sqrt{x}$ and the interval is $[0, 4] .$

Step 2. Find the exact area.

To find the area, we will use the formula of surface area as a definite integral which is $S = 2 蟺 {鈭�}_{a}^{b} f (x) \sqrt{1 + {(f' (x))}^{2}} d x .$

So,

$S = 2 蟺 {鈭�}_{0}^{4} \sqrt{x} \sqrt{1 + {(\frac{1}{2 \sqrt{x}})}^{2}} d x S = 2 蟺 {鈭�}_{0}^{4} \sqrt{x} \sqrt{1 + \frac{1}{4 x}} d x S = 蟺 {鈭�}_{0}^{4} \sqrt{1 + 4 x} d x S = 蟺 {[\frac{2}{3} {(1 + 4 x)}^{\frac{3}{2}} 路 \frac{1}{4}]}_{0}^{4} S = \frac{蟺}{6} [{(17)}^{\frac{3}{2}} - 1] S = \frac{蟺}{6} (17 \sqrt{17} - 1)$

Thus, the exact area is $S = \frac{蟺}{6} (17 \sqrt{17} - 1) .$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Exercises 63鈥�72, set up and solve a definite integral to find the exact area of each surface of revolution obtained by revolving the curve y = f(x) around the x-axis on the interval [a, b].
$f (x) = \sqrt{x}, [a, b] = [0, 4]$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the exact area.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Calculus

Decision Maths

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 63鈥�72, set up and solve a definite integral to find the exact area of each surface of revolution obtained by revolving the curve y = f(x) around the x-axis on the interval [a, b].f(x)=x,[a,b]=[0,4]

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the exact area.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Calculus

Decision Maths

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Exercises 63鈥�72, set up and solve a definite integral to find the exact area of each surface of revolution obtained by revolving the curve y = f(x) around the x-axis on the interval [a, b].
$f (x) = \sqrt{x}, [a, b] = [0, 4]$