More about volumes

Lesson 18
More about volumes

• Described how a plane region rotated about an
axis describes a solid.
• Found volumes of solids of revolution using
b
( ) 2 dh
" ! r(h) = !r(h) 2 " dh
a
Last time
• Determined the variable as given by the axis of
rotation.
b
a

Last time group work
Find the volume resulting when the curve y = 3x, on 1 ≤ x ≤ 3, is
rotated…
about the x-axis.
V = ! ( r(x) ) 2
3
" dx = ! ( 3x ) 2 3
" dx = ! 9x 2 3
" dx
1
3
3
9x
= !
3 1
= 3! x 3
3
1
1
= 3!(3 3 #1 3 ) = 78! $ 245units 3
1

Find the volume resulting when the curve y = 3x, on 1 ≤ x ≤ 3, is
rotated…
about the y-axis.
Last time group work
"
( ) 2
V = ! r(y)
= !
9
3
9
y 3
9
3 3
= !
27
dy
y 3
= !
9
3
"
9
3
# y
%
$ 3
&
2
( dy = !
'
"
3
9
y
9
2
dy
= !
9
= !
27 (93 ) 3 3 ) = 26! * 81.7units 3
"
3
9
y 2 dy

Few more examples of solids of revolution
Example 1: Find the solid generated by revolving the region below
the graph of
f (x) = 0.5 + 0.2sin(12x) between x = 0.5 and x = 2.5.
f (x) = 0.5 + 0.2sin(12x)
#
( )
2.5 2
0.5
The solid
V = ! 0.5 + 0.2sin(12 x) dx " 1.74 cubic units
Use a numerical technique like Simpson’s rule

Few more examples of solids of revolution
Example 2: Find the solid resulting from rotating
for 0 ≤ x ≤ 0.5, about the z-axis.
z = !2x +1
z = !2x +1
The solid
2
1 1
# 1" z $ 1
2 2 3
! ! 1( 1 2 ) ! % 1 & ! .
0 ' ( 0 4 4 ) 3 * 0 12
- -
V = dz = " z + z dz = z " z + z =
+ 2 ,

1. Start with the picture. For solids of revolution, sketch the region
involved first, then sketch the solid.
2. Make sure that everything within the integral is in terms of the
correct variable.
The axis you rotate about gives the variable to use for a solid of
revolution. When in doubt, refer to your diagrams.
3. Simplify the integral.
When working on volume problems:
4. Integrate (find the antiderivative)
5. Evaluate using the fundamental theorem.

Recall…
Volume of a blob:
Find cross sectional area at
height h: A(h)
Volume =
General Volumes
When the cross sections happen to be circles, we can use a
simplified formula using the radius of each circle.
But we cannot use this if the cross sections are not circles.

Example
Find the volume of a cylinder that has an elliptical base
with semi-major axis 5, semi-minor axis 4, and height 7.
Area of ellipse:
πab = π(4)(5) = 20π
So A(z) = 20π, and
volume =
Minor Axis
Major Axis

Examples for Practice
examples 1 and 2 on pp. 2-3 in your notes from Lesson 18.
Example 1: The Great Pyramid of Egypt is 410 feet high. Its base
is a square 755 feet by 755 feet.
Goal: find the volume of this solid.
Need: find the cross-sectional area at any height h.
(cross-sections are squares, we need to know the length of the side
of the pyramid at each height h.

Example 1
Great pyramid: square base, 755 ft by 755 ft and height 410 ft
a) Show
b) Area of square cross section at height h:
similar triangles

c) Integral for volume:
d)
Example 1
d) Compare your answer to the geometric formula a for the volume of a
square pyramid of height h with base length s:
V = 1
3 s2h = 1
3 (755)2 (410) = 77903416.67
.

Example 2
Set up an integral to find the volume of a half grapefruit whose diameter
is 5 inches. (you may assume that the whole grapefruit is a sphere.)
Each slice is a circle. Radius is given by x coordinate on
a semi-circle, radius 2.5.
Each slice is a circle. Radius is x

Example 2
What if we cut off the top; what is the volume from
height 0 to height 2?
Only change: limits from z = 0 to z = 2:

Example 3
Using Calculus: What is the volume of a waffle ice cream cone
whose diameter is 3.5 inches and height is 7 inches?
To find r, we need the equation of the side of the cone.
Two points on the side of the
cone are (0, 0) and (1.75, 7)
Equation of line: y = mx + b
With this example: y =
7
x = 4x
1.75
How does this help us? Vol = !(r(h)) 2 b
" dh
a
What is r ?
2 3
7
7 7
2 y ! 2 ! y
3
b
" #
Vol = * ! ( r( h)) dh = ! dy y dy 22.45 units
a * & ' = = $ %
0 4 16 * 0
( )
16 3
0

Example 4
The integral represents the volume of either a hemisphere
or a cone, and the variable of integration measures a
length.
!(81" h 2 #
9
) dh
0
a) Which shape is represented?
!
Vol = A(h) dh
a
b
Since this shape is a hemisphere.
!(81" h 2 ) = ! r 2
b) Give the radius of the hemisphere OR the radius and
height of the cone.
81! h 2 = r 2 " r 2 + h 2 = 81 = 9 2
The radius is 9 units.