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Chapter 5: Applications of Integration
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Section 5.2: Volume of a Solid of Revolution
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Example 5.2.7


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If $A$ is the plane region bounded by the $x$axis and the graphs of $y\={x}^{2}$ and $x\=1$, use the method of shells to calculate the volume of the solid of revolution formed when $A$ is rotated about the $y$axis.



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Solution



Mathematical Solution


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The height of the generic cylindrical shell for this solid is $L\left(x\right)\={x}^{2}$. This shell is formed from a rectangular slab $\mathrm{dx}$ thick, and of height $L\left(x\right)$ and width $2\mathrm{pi;}\mathrm{rho;}$, where $\mathrm{\ρ}\=x$.${}$

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Thus, the volume of the slab from which the shell is constructed is $2\mathrm{pi;}\mathrm{rho;}L\mathrm{dx}equals;2\mathrm{pi;}x\cdot {x}^{2}equals;2\mathrm{pi;}{x}^{3}\mathrm{dx}$.

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The total volume is then $2\mathrm{pi;}{\int}_{0}^{1}{x}^{3}\mathit{DifferentialD;}x$ = $\frac{{\mathrm{\pi}}}{{2}}$

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Maple Solution


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Figure 5.2.7(a) shows the
tutor defaulting to the shell method when finding the volume of the solid of revolution formed by rotating the plane region $A$ about a vertical axis.
In the section "Line of Revolution" select "Vertical" and leave the default 0 as the distance of the line of rotation from the coordinate axis. Click "Plot Options" and choose both Constrained Scaling and framed axes. Click the "Display" button to update the figure and to obtain the appropriate calculation.
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Figure 5.2.7(b) shows the solid segmented into a stack of concentric shells. The VolumeOfRevolution command at the bottom of the tutor, or the tutor itself can be used to generate a similar figure. (The actual code used to generate Figure 5.2.7(b) is hidden behind the table cell in which the figure resides.)
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Figure 5.2.7(a) Volume of Revolution tutor




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Student:Calculus1:VolumeOfRevolution(x^2, 0 .. 1, 'axis' = vertical, 'distancefromaxis' = 0, 'showvolume' = false, 'showsum' = true, 'showregion' = false, 'method' = midpoint, 'partition' = 6, 'output' = plot, axes = frame, sumvolumeoptions = [color = yellow, transparency = 0, lightmodel = none, style = patch], caption = "", labels = [x, z, y], tickmarks = [2, [0], [0, 1]], orientation = [125, 35, 0], glossiness = 0);


Figure 5.2.7(b) Solid segmented into shells




Calculation of the volume

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Expression palette: Definiteintegral template

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Context Panel: Evaluate and Display Inline


$2\mathrm{\pi}{\int}_{0}^{1}{x}^{3}\mathit{DifferentialD;}x$ = $\frac{{\mathrm{\pi}}}{{2}}$${}$



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