Math 2015
Lesson 16
Volume of a Tree Trunk
We begin working on finding the volumes of objects, starting with the simple shape of
the trunk of a tree.
Volumes
We will find the volume of a tree trunk by treating it as a
bunch of stacked cylinders, like the picture you see to the
right. We will find the volume of each cylinder and add them
up to get the total volume.
To start with we need to know that the volume of a cylinder
with radius r and thickness ∆h is just the area of the base
times the thickness, or V = (π r 2 ) Δh . (I choose ∆h as the
thickness, since it represents an increment of height.)
Exercise: Find the volume of a cylinder with radius 2 and
thickness 3.
Now we are ready to tackle finding the volume of a tree trunk in cubic feet. Below is a
table of values for the diameter of the trunk at different heights:
height
(feet)
diameter
(inches)
0
5
10
15
20
25
30
35
33.5
27.4
23
18.2
14.4
10.2
5.7
4
With your group, work through the three steps in Part 1 below.
Part 1:
Let’s find the volume of the first slice, from 0 to 5 feet.
•
What is the radius at the top and at the bottom of the slice, in feet?
•
What is the cross-sectional area at the top and at the bottom? Find your
answer in square feet.
•
So what is our estimate of the volume of the slice if we use the top radius?
The bottom radius?
79
Math 2015
Part 2:
•
Slice
Lesson 16
Let’s continue, using either the top or the bottom radius for each slice. (Your
group will be asked to use either the top or the bottom radius.)
Fill in the table giving the radius (either top or bottom) for each slice, the crosssectional area, and the volume of the slice:
Top
Bottom
Radius (ft)
Cross-sectional
Area (ft2)
Slice
1
1
2
2
3
3
4
4
5
5
6
6
7
7
Radius (ft)
Crosssectional
Area (ft2)
For example, if you were using the radius at the bottom of each slice, you would give
the radius at the bottom of the first slice as 1.396, and the first row of your table
would be as follows:
•
Slice
Radius (ft)
(bottom)
Cross-sectional
Area (ft2)
1
1.396
6.121
To find an estimate for the total volume of the tree using the radii, recall our LHS
and RHS from Lesson 1. We have Δh = 5 .
Volume Estimate, from top radii:
Volume Estimate, from bottom radii:
Part 3:
Do you think that either of the answers we obtained is necessarily an
overestimate or an underestimate? Why or why not?
How could we get a better estimate for the volume? (Come up with as many
different ideas as you can.)
80
Math 2015
Lesson 16
Volumes by Integration
In the above, we recognize that we are really using a Riemann sum to approximate the
volume. In this case, we know that: (2 things)
We know that our approximations should get better and better as the number of slices
goes to infinity (and therefore the thickness of each slice, ∆x, goes to zero). Therefore,
the volume of part of the tree trunk should be given by
where
•
h represents a height on the tree;
•
h = a is the height we wish to start at and h = b is the height we wish to stop at;
•
A(h) is a function that gives the cross-sectional area at height h.
We know of course that if the cross-sections are circles, and the cross-section at height h
has radius r(h), then the area is in fact A(h) = π (r(h))2, so we could also write the integral
as
Example:
Suppose we have determined that a tree has diameter at height h of
38.7
d(h) = 38.7h −1 2 =
inches.
h
Write down an integral which will give the total volume, in cubic feet, of the section of
tree trunk between heights h = 4 feet and h = 90 feet, and then calculate or approximate
that integral.
What we need is the area of each cross sectional slice of the tree. We assume each crosssectional slice is a circle, so that the area is still given by π (r(h))2. Since we are given
diameter instead of radius, we must find r(h):
But this will still not do, since the radius above is given in inches while the heights are
given in feet. If we want our final answer to come out in cubic feet, we must adjust the
units. (If we integrate π r 2, which is in inches, with respect to h, which is in feet, our
units on the integral will come out as in2-feet, which is nonsense.) So the radius in feet is
81
Math 2015
Lesson 16
Now we’re ready! We need to integrate the cross-sectional area from 4 to 90.
Summary
Today, we have
•
Determined a method for finding the volume of a tree trunk using Riemann sums.
•
Recognized our method of finding volumes as an integral, and determined that to
find the volume of a solid, we simply integrate the cross-sectional area of the
solid.
82