Johann Peter Gustav Lejeune Dirichlet1805 – 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 strip and revolve it about the y-axiswe get a hollow cylinder.

cross section

If we add all of the cylinders together, we can reconstruct the original object to obtain its volume.

2 1y x

x

( ) b

aV A x dx

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.

2x

cross section

The volume of a thin, hollow cylinder is given by:

Lateral surface area of cylinder thicknessV

=2 r h dx r is the x value of the function.circumference height thickness

h is the y value of the function.thickness is dx.

2=2 1 x x dx

r h thicknesscircumference

2 1y x

x

( ) b

aV A x dx

Example

cross section

=2 thicknessr h

2=2 1 x x dx

r h thicknesscircumference

Now we bring out the Super Sum, the Great Accumulator

2 2

02 1 x x dx

2 3

02 x x dx

24 2

0

1 124 2x x

2 4 2

312 units

This is called the Shell method because we use cylindrical shells.

2 1y x

x

Example

Example

24( ) 10 169

f x x x

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.

2 2( )d

O icV r r dy

24( ) 10 169

f x x x Shell method:

Lateral surface area of cylinder

( ) = circumference heightA x

( ) = 2A x r h

Volume of the shell 2 ( )x f x dx

If we take a vertical sliceand revolve it about the y-axiswe get a cylinder.

x

( )f x

( )b

aV A x dx

( ) = 2 ( )A x x f x

Example

24( ) 10 169

f x x x Volume of thin cylinder 2 ( ) x f x dx

8 2

2

42 10 16 9

x x x dx r

h thickness

3160 cm3502.655 cm

circumference

( )f xVolume of thin cylinder 2 r h dx

x

A(x)( )

b

aV A x dx

Example

Example

2

Example:3Consider the bounded region determined by , , and the axis.

1Determine the following.a) Find the point of intersection.b) Find the area of the bounded region.c) The region revolv

xy y e yx

ed around the line 4.d) The region revolved around the line 1.e) Planes perpendicular to the -axis intersect the solid in isosceles right triangles where the hypotenuse is in the base.

xy

x

(0.700, 2.014)

2

31

yx

xy ey

x