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Calculus

Volume of Revolution

Volume of Revolution - Shell Method

         

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.

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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 \(\alpha?\)

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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 \(\alpha?\)

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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 \(\alpha?\)

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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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