Question

One way to think of this volume is as an accumulation of disks as $x$varies from $1to3$, as shown next at the left. The disk at a given $x\in [1,3]$has radius$r=f\left(x\right)$and thus area $\pi (f{\left(x\right))}^2$. As we will see in Section $6.1$, the definite integral

role="math" localid="1650809851583" $\mathrm{\pi }{\int }_1^3{\left(f\left(x\right)\right)}^{2}dx$

calculates the volume of the solid. Use integration by parts to calculate this volume.

Short answer

The volume of the solid is, $1.029\mathrm{\pi }$cubic units.

Step-by-step solution

Step 1. Given information

$\mathrm{\pi }{\int }_1^3{\left(f\left(x\right)\right)}^{2}dx$.

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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,

$$\begin{gathered} r=\ln x \\ f\left(x\right)=\ln x \end{gathered}$$

So, $\mathrm{\pi }{\int }_1^3{\left(f\left(x\right)\right)}^{2}dx=\mathrm{\pi }{\int }_1^3{\left(\ln x\right)}^{2}dx$

Let us evaluate the obtained integral.

$\mathrm{\pi }{\int }_1^3{\left(\ln \mathrm{x}\right)}^{2}dx=\mathrm{\pi }{\int }_1^3{\left(\ln x\right)}^{2}dx$

By using the Integrating by parts:

$$\begin{gathered} \int fg\text{'}=fg-\int f\text{'}g \\ f\text{'}=\frac{2\ln x}{x},g\text{'}=1 \\ \mathrm{\pi }{\int }_1^3{\left(\ln \mathrm{x}\right)}^{2}dx=\mathrm{\pi }{\left[{\mathrm{xln}}^2\mathrm{x}-\int 2\ln \mathrm{xdx}\right]}_1^3 \\ =\mathrm{\pi }{\left[{\mathrm{xln}}^2\mathrm{x}-2\int \ln \mathrm{xdx}\right]}_1^3 \\ =\mathrm{\pi }{\left[{\mathrm{xln}}^2\mathrm{x}-2\left[\mathrm{x}\ln \mathrm{x}-\mathrm{x}\right]\right]}_1^3 \\ =\mathrm{\pi }{\left[{\mathrm{xln}}^2\mathrm{x}-2\mathrm{xln}\mathrm{x}+2\mathrm{x}\right]}_1^3 \\ =\mathrm{\pi }\left[(3{\ln }^{2}3-6\ln 3+6)-({\ln }^{2}1-2\ln 1+2)\right] \\ =\mathrm{\pi }\left[\left(3.621-6.592+6\right)-2\right] \\ =1.029\mathrm{\pi }\mathrm{cubic}\mathrm{units} \end{gathered}$$