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True or False In using the shell method to find the volume of a solid revolved about the \(x\)-axis, the integration occurs with respect to \(x\).

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True or False The volume of a cylindrical shell of outer radius \(R\), inner radius \(r\), and height \(h\) is given by \(V=\pi r^{2}h\).

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True or False The volume of a solid of revolution can be found using either washers or cylindrical shells only if the region is revolved about the \(y\)-axis.

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True or False If \(y=f(x)\) is a function that is continuous and nonnegative on the closed interval \([a, b], a\geq 0\), then the volume \(V\) of the solid generated by revolving the region bounded by the graph of \(f\) and the \(x\)-axis from \(x=a\) to \(x=b\) about the \(y\)-axis is \(V=2\pi \int_{a}^{b}xf(x)~{\it dx}\).

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\(y=x^{2}+1\), the \(x\)-axis, \(0\leq x\leq 1\); about the \(y\)-axis

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\(y=x^{3}, y=x^{2}\); about the \(y\)-axis

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\(y=\sqrt{x}, y=x^{2}\); about the \(y\)-axis

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\(y=\dfrac{1}{x}\), the \(x\)-axis, \(x=1, x=4\); about the \(y\)-axis

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\(y=x^{3}\), the \(y\)-axis, \(y=8\); about the \(x\)-axis

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\(y=\sqrt{x}\), the \(y\)-axis, \(y=2\); about the \(x\)-axis

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\(x=\sqrt{y}\), the \(y\)-axis, \(y=1\); about the \(x\)-axis

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\(x=4\sqrt{y}\), the \(y\)-axis, \(y=4\); about the \(x\)-axis

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\(y=x, y=x^{2}\); about the \(x\)-axis

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\(y=x, y=x^{3}\); in the first quadrant; about the \(x\)-axis

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\(y=e^{-x^{2}}\), and the \(x\)-axis, from \(x=0\) to \(x=2\); about the \(y\)-axis

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\(y=x^{3}+x\) and the \(x\)-axis, \(0\leq x\leq 1;\) about the \(y\)-axis

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\(y=x^{2}, y=4x-x^{2}\); about \(x=4\)

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\(y=x^{2}, y=4x-x^{2}\); about \(x=3\)

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\(y=x^{2}, y=0, x=1, x=2\); about \(x=1\)

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\(y=x^{2}, y=0, x=1, x=2\); about \(x=-2\)

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\(x=y-y^{2}\), the \(y\)-axis; about \(y=-1\)

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\(x=y-y^{2}\), the \(y\)-axis; about \(y=1\)

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\(y=\sqrt[3]{x}\), the \(y\)-axis, \(y=2\); about the \(x\)-axis

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\(y=\sqrt{x}+x\), the \(x\)-axis, \(x=1, x=4\); about the \(y\)-axis

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\(y=x^{3}\) and \(y=x\) to the right of \(x=0\); about the \(y\)-axis

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\(y=x^{3}, y=x^{2}\); about the \(x\)-axis

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\(y=3x^{2}\) and \(y=30-x\) to the right of \(x=1\); about the \(y\)-axis

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\(y=3x^{2}\) and \(y=30-x\) to the right of \(x=1\); about the \(x\)-axis

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\(y=x^{2}\) and \(y=8-x^{2}\) to the right of \(x=1\); about the \(x\)-axis

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\(y=x^{2}\) and \(y=8-x^{2}\) to the right of \(x=1\); about the \(y\)-axis

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\(y=x^{2}, y=x\); about the \(y\)-axis

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\(y=x^{2/3}+x^{1/3}\), the \(x\)-axis, \(x=1, x=8\); about the \(y\)-axis

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\(y=\sqrt{x}\) and \(y=18-x^{2}\) to the right of \(x=1\); about the \(y\)-axis

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\(y=\sqrt{x}\) and \(y=18-x^{2}\) to the right of \(x=1\); about the \(x\)-axis

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Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=\dfrac{1}{(x^2+1)^2}\) and the \(x\)-axis from \(x=0\) to \(x=1\) about the \(y\)-axis as shown in the figure.

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Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=\sqrt{x+1}+x\) and the \(x\)-axis from \(x=0\) to \(x=3\) about the \(y\)-axis.

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Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region in the first quadrant bounded by the \(x\)-axis and the graph of \(y=2x-x^{2}\):

1. about the \(x\)-axis.
2. about the \(y\)-axis.
3. about the line \(x=3\).
4. about the line \(y=1\).

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Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region in the first quadrant bounded by the \(x\)-axis and the graph of \(y=x\sqrt{9-x}\).

1. about the \(x\)-axis.
2. about the \(y\)-axis.
3. about the line \(y=-3\).
4. about the line \(x=-2\).

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Volume of a Solid of Revolution Suppose \(f(x)\geq 0\) for \(x\geq 0\), and the region bounded by the graph of \(f\) and the \(x\)-axis from \(x=0\) to \(x=k, k > 0\), is revolved about the \(x\)-axis. If the volume of the resulting solid is \(\dfrac{1}{5}k^{5}+k^{4}+\dfrac{4}{3}k^{3}\), find \(f\).

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Volume of a Solid of Revolution Find the volume of the solid generated by revolving about the \(y\)-axis the region bounded by the graph of \(y=e^{-x^{2}}\) and the \(y\)-axis from \(x=0\) to the positive \(x\)-coordinate of the point of inflection of \(y=e^{-x^{2}}\).

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Volume of a Solid of Revolution Show that the volume \(V\) of the solid generated by revolving the region bounded by the graph of \(y=\sin ^{3}x\) and the \(x\)-axis from \(x=0\) to \(x=\pi\) about the line \(x=-\pi\) is given by \(V=4\pi^{2}\).

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Comparing Volume Formulas In the figure below, the region bounded by the graphs of \(y=g(x), y=f(x)\), and the \(y\)-axis is to be revolved about the \(y\)-axis. Show that the resulting volume \(V\) is given by:

1. \(V=\pi a^{2}[f(a) -g(a)] +\,2\pi \displaystyle \int_{a}^{b}\!\!x[f(x)-g(x)]\, {\it dx}\) if the shell method is used.
2. \(V=\pi \displaystyle \int_{g(a)}^{g(b)}[g^{-1}(y)]^{2}{\it dy}\,{+}\,\pi \int_{g(b)}^{f(a)} [f ^{-1}(y)] ^{2}{\it dy}\) if the disk method is used

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Suppose a plane region of area \(A\) that lies to the right of the \(y\)-axis is revolved about the \(y\)-axis, generating a solid of volume \(V\). If this same region is revolved about the line \(x=-k, k > 0\), show that the solid generated has volume \(V+2\pi kA\).

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Find the volume generated by revolving the region bounded by the parabola \(y^{2}=8x\) and its latus rectum about the latus rectum:

1. by using the disk method.
2. by using the shell method.

(The latus rectum is the chord through the focus perpendicular to the axis of the parabola.) See the figure.