1. Volumes by Cylindrical Shells r

2. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells The method applies to solids obtained by letting the domain under the graph of a function rotate about the vertical axis.

3. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindres The volume of a cylinder of height h on a disk of radius R is V = height area of the bottom V = π R 2 h h R

4. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells A Cylindrical Shell is obtained by removing, from a solid cylinder, a cylinder of smaller radius as indicted in the picture. h rR r Volume of the Cylindrical Shell = π R 2 h - π r 2 h = π h (R 2 - r 2 ) = π h (R - r)(R + r).

5. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells h rR r Area of the Rectanlge A = (R - r)h. Volume of the Shell= π h(R - r)(R + r) =. h R r A Cylindrical Shell can also be obtained by letting the green rectangle in the picture rotate about the vertical axis.

6. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells When all the rectangles of a Riemann sum rotate about the vertical axis we get a Cylindrical Shell Approximation of the Volume of the solid obtained by letting the domain bounded by the graph of the given function rotate about the vertical axis. Volume of the Shell=.

7. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shell Approximations f Let x k be the midpoint of a subinterval. Rectangle of height f(x k ) and width Δx has the area f(x k ) Δx. The rectangle rotates about the vertical axis forming a cylindrical shell of volume V k = 2 π x k f(x k ) Δx assuming that x k > 0.

8. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells When all the rectangles of a Riemann sum rotate about the vertical axis we get a Cylindrical Shell Approximation of the Volume. Valid if 0 a b and if f is non-negative on the interval [a,b].

9. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells Formula The Volume of a Solid obtained by letting the domain bounded by the graph of a function f rotate about the vertical axis: Here we assume the (a,b) is either an interval in the positive real axis or in the negative real axis. I.e. we assume that 0 (a,b).

10. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Example

11. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Example

12. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells Formula The Volume of a Solid obtained by letting the domain bounded by the graph of a function f rotate about the vertical axis: Here we assume the (a,b) is either an interval in the positive real axis or in the negative real axis. I.e. we assume that 0 (a,b).