Q. 1 One way to think of this volume ... [FREE SOLUTION]

Chapter 5: Q. 1 (page 431)

One way to think of this volume is as an accumulation of disks as $x$ varies from $1 t o 3$ , as shown next at the left. The disk at a given $x 鈭� [1, 3]$ has radius $r = f (x)$ and thus area $蟺 (f {(x))}^{2}$ . As we will see in Section $6.1$ , the definite integral
role="math" localid="1650809851583" $蟺 {鈭�}_{1}^{3} {(f (x))}^{2} d x$
calculates the volume of the solid. Use integration by parts to calculate this volume.

Short Answer

Expert verified

The volume of the solid is, $1.029 蟺$ cubic units.

Step by step solution

Step 1. Given information

$蟺 {鈭�}_{1}^{3} {(f (x))}^{2} d x$ .

Step 2. 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font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="2.5" y="16">r</text><text font-family="math17f39f8317fbdb1988ef4c628eb" font-size="16" text-anchor="middle" x="14.5" y="16">=</text><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="26.5" y="16">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="35.5" y="16">(</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="48.5" y="16">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="41.5" y="16">x</text></svg>" role="math" localid="1650809798741" r=fx.

Therefore,

$r = \ln x f (x) = \ln x$

So, $蟺 {鈭�}_{1}^{3} {(f (x))}^{2} d x = 蟺 {鈭�}_{1}^{3} {(\ln x)}^{2} d x$

Let us evaluate the obtained integral.

$蟺 {鈭�}_{1}^{3} {(\ln x)}^{2} d x = 蟺 {鈭�}_{1}^{3} {(\ln x)}^{2} d x$

By using the Integrating by parts:

$鈭� f g' = f g - 鈭� f' g f' = \frac{2 \ln x}{x}, g' = 1 蟺 {鈭�}_{1}^{3} {(\ln x)}^{2} d x = 蟺 {[{xln}^{2} x - 鈭� 2 \ln xdx]}_{1}^{3} = 蟺 {[{xln}^{2} x - 2 鈭� \ln xdx]}_{1}^{3} = 蟺 {[{xln}^{2} x - 2 [x \ln x - x]]}_{1}^{3} = 蟺 {[{xln}^{2} x - 2 xln x + 2 x]}_{1}^{3} = 蟺 [(3 \ln^{2} 3 - 6 \ln 3 + 6) - (\ln^{2} 1 - 2 \ln 1 + 2)] = 蟺 [(3.621 - 6.592 + 6) - 2] = 1.029 蟺 cubic units$

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Short Answer

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