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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