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Volume of Revolution

Integration extends to the third dimension with the means to compute the volumes of shapes obtained by revolving regions around an axis, such as spheres, toroids, and paraboloids.

Shell Method - Basic

The above diagram shows the points \(R=(7,\frac{1}{7})\) and \(P=(14,\frac{1}{14}),\) which lie on the curve \(y=\frac{1}{x},\) point \(Q = (0, \frac{1}{7}) \) on the \(y\)-axis, and points \(A = (7,0)\) and \(B = (14,0)\) on the \(x\)-axis. If \(S\) is the solid obtained by rotating the region \(R\) bounded by \[\overline{OQ}, \overline{QR}, \widehat{RP}, \overline{PB}, \text{ and } \overline{OB}\] about the axis of rotation \(x=-7\) and the volume of \(S\) is \(\alpha \pi, \) what is \(\alpha?\)

Note: The above diagram is not drawn to scale.

Let \(S\) be the solid obtained by rotating the region \(R\) bounded by \( y = x^2\) and \( y = \sqrt{x}\) about the line \(x=-10 \). If the volume of \(S\) is \(\alpha \pi, \) what is \(\alpha?\)

Let \(S\) be the solid obtained by rotating the region \(R\) bounded by \( y = x-9, x = 15,\) and the \(x\)-axis about the axis of rotation \(x=19 .\) If the volume of \(S\) is \(\alpha \pi, \) what is \(\alpha?\)

Let \(S\) be the solid obtained by rotating the region \(R\) bounded by \(y = x^3, x = 1, x = 5,\) and the \(x\)-axis about the line \(x=8 \). If \(10\) times the volume of \(S\) is \(\alpha \pi ,\) what is \(\alpha?\)

Let \( R\) be the region bounded by \( y = 3x+2 , x = 4\), and the x-axis. Let \(S\) be the solid obtained by rotating \(R\) about the axis \(x=0 \). The volume of \(S\) has the form \(T \pi \). What is the value of \(T\)?

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