Volume: disk/washer method:   Lecture Video 1   Lecture Video 2

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

  1. Find the volume of the solid obtained by rotating the region bounded by \(y=\sqrt{x}, \: x=4 \) and \(x\)-axis about the \(x\)-axis.  solution

  2. Find the volume of the solid obtained by rotating the region bounded by \(y=\sqrt{x}\) and \(y=x\) about the \(x\)-axis. solution

  3. Sketch the region enclosed by the graphs of \(y=x^2\) and \(y=3x\). Now use a definite integral to find the volume of revolution obtained by rotating the region about \(x\)-axis.  solution

  4. Find the volume of the solid of rotation of the region enclosed by the curves \(y=x, \: y=x+2, \: x=0, \: x=2\) about the \(x\)-axis. solution

  5. Find the volume of the solid of rotation of the region enclosed by the curves \(y=2+e^x, \: y=1, \: x=0, \: x=2\) about the line \(y=0\). solution

  6. Find the volume of the solid obtained by rotating the region bounded by the curves: \( y=\sqrt{x+2}\) and \( y=x\) about the x-axis. solution


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