Chapter 6: Q. 13 (page 575)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information

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

Step 2. differentiate given function

$f (x) = \sin (πx) f' (x) = πcos (πx)$

The required surface area is given by

$S = 4 π \int_{0}^{1} f (x) \sqrt{1 + {(f' (x))}^{2}} dx = 4 π \int_{0}^{1} \sin (πx) \sqrt{1 + π^{2} \cos^{2} (πx)} dx u = πcos (πx) \sin (πx) dx = \frac{1}{π^{2}} du S = 4 π \int_{π}^{- π} \sqrt{1 + u^{2}} \cdot (- \frac{1}{π^{2}}) du = 4 π \int_{π}^{- π} \sqrt{1 + u^{2} du} = - \frac{4}{π} {[\frac{u}{2} \sqrt{1 + u^{2}} + \frac{1}{2} \ln (|u + \sqrt{1 + u^{2}}|)]}_{π}^{- π} = - \frac{2}{π} [- π \sqrt{1 + π^{2}} + \ln (|- π + \sqrt{1 + π^{2}} - π \sqrt{1 + π^{2}} - \ln (|π + \sqrt{1 + π^{2}}|)|)] = \ln (x) - \ln (y) = \ln (\frac{x}{y}) S = \frac{2}{π} [2 π \sqrt{1 + π^{2}} + \ln (|\frac{\sqrt{1 + π^{2}} + π}{\sqrt{1 + π^{2}} - π}|)] = 4 \sqrt{1 + π^{2}} + \frac{4}{π} \ln (π + \sqrt{1 + π^{2}})$

Step 3. The solution

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

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The area of the surface obtained by revolving the curve
$f (x) = \sin (π x)$ 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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The area of the surface obtained by revolving the curvefx=sinπxaround 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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Applied Mathematics

Statistics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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