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

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  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 \(y=x^3, y=8\) and the y-axis about the y-axis. solution

  7. Find the volume of the solid generated by rotating the region enclosed by the curves \( y=x\) and \( y=x^2\) about the y-axis. solution

  8. What is the volume of the solid generated by rotating the region bounded by the curves \( y^2=x \) and \( y=2-x\) about the y-axis? solution

  9. 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

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

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

  12. 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 

  13. Find the volume of the solid obtained by rotating the region bounded by the curves \( y=8x^3, \, y=0, \,x=1 \) about the line \(x=2\).  solution