Q. 13 The area of the surface obtained... [FREE SOLUTION]

Chapter 6: Q. 13 (page 575)

The area of the surface obtained by revolving the curve
$f (x) = \sin (蟺 x)$ around the x-axis on $[- 1, 1]$ .

Short Answer

Expert verified

The area of surface obtained by revolving the given curve is $4 \sqrt{1 + 蟺^{2}} + \frac{4}{蟺} \ln (\sqrt{1 + 蟺^{2}} + 蟺)$

Step by step solution

Step 1. Given information

The given curve is $f (x) = \sin (蟺虫)$ around the x-axis $[- 1, 1]$

Step 2. differentiate given function

$f (x) = \sin (蟺虫) f' (x) = 蟺肠辞蝉 (蟺虫)$

The required surface area is given by

$S = 4 蟺 {鈭�}_{0}^{1} f (x) \sqrt{1 + {(f' (x))}^{2}} dx = 4 蟺 {鈭�}_{0}^{1} \sin (蟺虫) \sqrt{1 + 蟺^{2} \cos^{2} (蟺虫)} dx u = 蟺肠辞蝉 (蟺虫) \sin (蟺虫) dx = \frac{1}{蟺^{2}} du S = 4 蟺 {鈭�}_{蟺}^{- 蟺} \sqrt{1 + u^{2}} 路 (- \frac{1}{蟺^{2}}) du = 4 蟺 {鈭�}_{蟺}^{- 蟺} \sqrt{1 + u^{2} du} = - \frac{4}{蟺} {[\frac{u}{2} \sqrt{1 + u^{2}} + \frac{1}{2} \ln (|u + \sqrt{1 + u^{2}}|)]}_{蟺}^{- 蟺} = - \frac{2}{蟺} [- 蟺 \sqrt{1 + 蟺^{2}} + \ln (|- 蟺 + \sqrt{1 + 蟺^{2}} - 蟺 \sqrt{1 + 蟺^{2}} - \ln (|蟺 + \sqrt{1 + 蟺^{2}}|)|)] = \ln (x) - \ln (y) = \ln (\frac{x}{y}) S = \frac{2}{蟺} [2 蟺 \sqrt{1 + 蟺^{2}} + \ln (|\frac{\sqrt{1 + 蟺^{2}} + 蟺}{\sqrt{1 + 蟺^{2}} - 蟺}|)] = 4 \sqrt{1 + 蟺^{2}} + \frac{4}{蟺} \ln (蟺 + \sqrt{1 + 蟺^{2}})$

Step 3. The solution

The area of surface obtained by revolving the graph of $f (x) = \sin (蟺虫)$ around x-axis on the interval $[- 1, 1]$ islocalid="1649139486946" $4 \sqrt{1 + 蟺^{2}} + \frac{4}{蟺} \ln (\sqrt{1 + 蟺^{2}} + 蟺)$

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The area of the surface obtained by revolving the curve
$f (x) = \sin (蟺 x)$ around the x-axis on $[- 1, 1]$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. differentiate given function

Step 3. The solution

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91影视

The area of the surface obtained by revolving the curvefx=sin蟺xaround the x-axis on-1,1.

Short Answer

Step by step solution

Step 1. Given information

Step 2. differentiate given function

Step 3. The solution

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Most popular questions from this chapter

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Applied Mathematics

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The area of the surface obtained by revolving the curve
$f (x) = \sin (蟺 x)$ around the x-axis on $[- 1, 1]$ .