Johann Peter Gustav Lejeune Dirichlet 1805 – 1859 Johann Peter Gustav Lejeune Dirichlet 1805 – 1859 Dirichlet proved the convergence of Fourier series, as well as developed the modern definition of a function, also he contributed to analytic number theory.

If we take a vertical stripand revolve it about the y-axis we get a hollow cylinder. cross section If we add all of the cylinders together, we can reconstruct the original object to obtain its volume. x Like always, we need to find A(x). We will return to this problem. Example Find the volume of the solid of revolution for the bounded region revolved about the y-axis.

cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx. x Example

cross section Now we bring out the Super Sum, the Great Accumulator This is called the Shell method because we use cylindrical shells. x Example

Find the volume generated when this shape is revolved about the y-axis. It’s not easy to solve for x, so we do not want to use a horizontal slice to find the volume.

Shell method: If we take a vertical slice and revolve it about the y-axis we get a cylinder. x Example

x A(x)A(x)