Q. 64 In Exercises 63鈥�72, set up and... [FREE SOLUTION]

Chapter 6: Q. 64 (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].
$f (x) = \frac{1}{x}, [a, b] = [1, 10]$

Short Answer

Expert verified

The exact area of the surface of the revolution obtained by revolving the curve $f (x) = \frac{1}{x}$ around the x-axis on the interval $[1, 10]$ is localid="1650696963685" $S = 蟺 [\sqrt{2} 鈭� \frac{\sqrt{1 + (100)^{2}}}{100} 鈭� \ln (1 + \sqrt{2}) + \ln (1 + \sqrt{1 + (100)^{2}})] .$

Step by step solution

Step 1. Given Information.

The given curve is $f (x) = \frac{1}{x}$ and the interval is $[1, 10] .$

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,

localid="1650696996195" $S = 2 蟺 {鈭�}_{1}^{10} \frac{1}{x} \sqrt{1 + {(鈭� \frac{1}{x^{2}})}^{2}} d x S = 2 蟺 {鈭�}_{1}^{10} \sqrt{1 + \frac{1}{x^{4}}} 鈰� x^{2} 鈰� \frac{1}{x^{3}} d x Now, let u = \frac{1}{x^{2}}, (- \frac{1}{2}) d u = (\frac{1}{x^{3}}) d x S = 2 蟺 {鈭�}_{1}^{1 / 100} \sqrt{1 + u^{2}} 鈰� \frac{1}{u} 鈰� (鈭� \frac{1}{2}) d u S = 鈭� 蟺 {鈭�}_{1}^{1 / 100} \frac{\sqrt{1 + u^{2}}}{u} d u S = 鈭� 蟺 {[\sqrt{1 + u^{2}} 鈭� \ln |\frac{1 + \sqrt{1 + u^{2}}}{u}|]}_{1}^{1 / 100} S = 鈭� 蟺 [\sqrt{1 + {(\frac{1}{100})}^{2}} 鈭� \ln |\frac{1 + \sqrt{1 + {(\frac{1}{100})}^{2}}}{(1 / 100)}| 鈭� \sqrt{2} + \ln (1 + \sqrt{2})] S = 蟺 [\sqrt{2} 鈭� \frac{\sqrt{1 + (100)^{2}}}{100} 鈭� \ln (1 + \sqrt{2}) + \ln (1 + \sqrt{1 + (100)^{2}})]$

Thus, the exact area islocalid="1650696984063" $S = 蟺 [\sqrt{2} 鈭� \frac{\sqrt{1 + (100)^{2}}}{100} 鈭� \ln (1 + \sqrt{2}) + \ln (1 + \sqrt{1 + (100)^{2}})] .$

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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].
$f (x) = \frac{1}{x}, [a, b] = [1, 10]$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the exact area.

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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)=1x,[a,b]=[1,10]

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

Calculus

Decision Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Pure 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].
$f (x) = \frac{1}{x}, [a, b] = [1, 10]$