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 \( x^2+y^2 -100 =0 \) 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 yellow-colored region in the above diagram is bounded by \[\begin{array} &y= 18\sin x - a \ (0 < a \ < 18), &x=0, &x=\pi, &y=0 .\end{array}\] What is the value of \(a\) that minimizes the volume of the solid obtained by rotating the region around the \(x\)-axis?

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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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