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