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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.
Let \(V\) be the volume of the solid obtained by revolving the curve \( y = x^2 \) from \( x = 2\) to \(3\) about the \(x\)-axis. What is \(10 V?\)
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Let \(V\) be the volume of the solid obtained by revolving the curve \( x^2+y^2 -100 =0 \) from \( x = 0\) to \(x=4\) about the \(x\)-axis. What is \(3 V? \)
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If \(a\) is a positive number such that the volume of the solid obtained by rotating the ellipse \( {x}^2 + a{y}^2 = 1\) around the \(x\)-axis is \( \frac{4}{57}\pi ,\) what is \(a?\)
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The volume of the solid obtained by rotating the region bounded by \( y = x^2 - 2x\) and \( y = x \) about the line \( y = 6\), has the form \( \frac {a}{b} \pi\), where \(a\) and \(b\) are positive coprime integers. What is the value of \(a+b\)?
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