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Wize University Calculus 2 Textbook > Applications of Integrals

Volumes of Revolution (Cylindrical Shells Method)

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Volumes of Revolution (Cylindrical Shells)

The question will ask you to compute the volume of the solid obtained by rotating a bouonded region around the x or y axis (or a line parallel to one of these axes).






The volume of a solid obtained by rotating
  • the graph of y=f(x)y=f(x) from x=ax=a to x=bx=b around the y-axis is given by V=∫x=ax=b2πx f(x) dx\displaystyle V=\int_{x=a}^{x=b}2\pi x\ f(x)\ dx
  • the graph of x=g(y)x=g(y) from y=cy=c to y=dy=d around the x-axis is given by V=∫y=cy=d2πy g(y) dy\displaystyle V=\int_{y=c}^{y=d}2\pi y\ g(y)\ dy

Wize Tip
If it's not clear what the outer and inner radius are, the method of using cylindrical shells will be easier than using disks/washers!

Practice: Volumes using Cylindrical Shells

Find the volume of the solid generated by rotating the region bounded by y=x2+1y=x^2+1 and the x-axis on the interval 0≤x≤20\le x \le2 around the y-axis.
Extra Practice
Find the volume of the solid generated by rotating the region bounded by y=x2+1y=x^2+1 and the x-axis on the interval 0≤x≤20\le x \le2 around the y-axis.
Practice: Volumes using Cylindrical Shells
Find the volume of the solid generated by rotating the region bounded by y=x2+1y=x^2+1 and the x-axis on the interval 0≤x≤20\le x \le2 around the y-axis.
Practice: Volumes using Cylindrical Shells
Find the volume of the solid generated by rotating the region bounded by y=x2+1y=x^2+1 and the x-axis on the interval 0≤x≤20\le x \le2 around the y-axis.
Practice: Volumes using Cylindrical Shells
Find the volume of the solid generated by rotating the region bounded by y=x2+1y=x^2+1 and the x-axis on the interval 0≤x≤20\le x \le2 around the y-axis.