Volume: Method of cylindrical shells

Watch this animation to get an idea of cylindrical shells:


  1. Find the volume of the solid generated by revolving the region bounded by the curves \( y=\sqrt{x}, \, \, x=2\) and \( y=0 \) about the y-axis.  solution

  2. Let R be the region bounded by the curves \(y=\sqrt[3]{x}, \, \, y=0\) and \( x=1\). Find the volume of the solid generated by revolving the region R about the y-axis.  solution

  3. Find the volume of the solid obtained by rotating about the line \( x = -2\), the region bounded by \(y=x\) and \( y=x^2\).   solution

  4. Find the volume of the solid obtained by rotating about the line \( x = -1 \), the region bounded by \(y=2x^2-x^3\) and \(y=0\).   solution