Volume of Revolution
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 \(T\)?