3. You will be able to calculate volumes
of irregular shaped solids
4. Some of the Professional fields that
are using this particular concepts of
Integral Calculus
MRI & CAT
scan
Construction
Idustrial
Designs
Containers
and Packaging
6. Make Sure You Remember Process for
Calculating Area
Divide the region into n pieces.
Approximate the area of each piece with a rectangle.
Add together the areas of the rectangles.
Take the limit as n goes to infinity.
The result gives a definite integral.
7. General Idea - Slicing
1.
Divide the solid into n pieces (slices).
2.
Approximate the volume of each slice.
3.
Add together the volumes of the slices.
4.
Take the limit as n goes to infinity.
5.
The result gives a definite integral.
9. Volume of a Slice
Volume of a cylinder?
V
r
2
r h
What if the ends are
not circles?
V
A
h
Ah
What if the ends are not
perpendicular to the side?
No difference!
(note: h is the distance
between the ends)
10. Volume of a Solid
x
A(xk)
Vslice
a
xk
n
V
lim
n
A( xk ) x
k 1
A( xk ) x
b
b
a
A( x)dx
The hard part?
Finding A(x).
11. Volumes by Slicing: Example
Find the volume of the solid of revolution formed by rotating the
x from
region bounded by the x-axis and the graph of y =
x=0 to x=1, about the x-axis.
12. Here is a Problem for You:
Find the volume of the solid of revolution formed by rotating the
region bounded by the x-axis and the graph of y = x4, from x=1 to
x=2, about the x-axis.
Ready?
A(x) = p(x4)2= px8.
14. • Consider the area between two functions
rotated about the axis
• Now we have a hollow solid
• We will sum the volumes of washers
15. Setting up the Equation
r
R
Outer
Function
Inner
Function
16. Solids of Revolution
A solid obtained by revolving a region around a line.
When the axis of rotation is
NOT a border of the region.
f(x)
g(x)
Creates a “pipe” and the
slice will be a washer.
Find the volume of the solid
and subtract the volume of
the hole.
V
V
b
b
2
f ( x) dx
a
b
a
f ( x)
2
a
2
g ( x) dx
a
xk
b
2
g ( x) dx
NOTE: Cross-section is
perpendicular to the
axis of rotation.
17. Example:
Find the volume of the solid formed by revolving the
region bounded by y = (x) and y = x² over the interval [0,
1] about the x – axis.
b
([ f ( x)]2 [ g ( x)]2 )dx
V
a
1
V
2
x
x
2 2
0
V = p ò (x - x )dx
1
4
0
æx x ö
3
V =pç - ÷ =
è 2 5 ø 0 10
2
5
1
dx
19. So……how do you calculate
volumes of revolution?
• Graph your functions to create the region.
• Spin the region about the appropriate axis.
• Set up your integral.
• Integrate the function.
• Evaluate the integral.
