Volume: Method of cylindrical shells

Watch this animation to get an idea of cylindrical shells (click me)


  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 the region bounded by the curves \(y=2x\) and \( y=x^2\) about the \(y\)-axis. solution

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

  5. 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

  6. 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

  7. Find the volume of the solid generated by revolving the region bounded by the curves \( y=\sqrt{x}, \,\, x=4\) and \(y=0 \) about the the line \(y=-2\).  solution

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