Volume Applications of Integrals
Instructor: Steven Krake
Class: MTH 252
Date: 12/07/2009
(Monsite)
By: Joshua Sackos and Zach Gunther
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Overview
Information:
Consider the region, where 𝑥
≥ 0, bounded by 𝑦 =
rotated about the line 𝑥 = 4.


𝑥3
8
and 𝑦 = 2𝑥. This region will be
Find the volume of the resulting solid by using the washer method.
Find the volume of the resulting solid by using the shell method.
Tasks:
Graph both of the functions on the same Cartesian graph, concerning the values of x which are
greater than or equal to zero, and then rotate the resulting curves about the vertical line at 𝑥 =
4. After rotating the functions a 3-dimensional object will be generated whose volume is to be
determined by using two different methods, the washer method and the shell method.
Solution:
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
Identify the domain of the graph that must be created. Given that 0 ≤ 𝑥 and that we
are rotating the functions about the vertical line at 𝑥 = 4, we can conclude that our
domain is 0 ≤ 𝑥 ≤ 8.
Make a table with (x, y) coordinates for each function evaluated for 0 ≤ 𝑥 ≤ 4.
Plot the resulting (x, y) coordinates on a color coded graph containing both functions on
it for 0 ≤ 𝑥 ≤ 4. For values of 4 ≤ 𝑥 ≤ 8 use symmetry about x = 4 to plot the
necessary points.
Shade the area bounded between the curves on the interval [0, 4] and [4, 8] to identify
the area of concern.
Identify where the functions intersect each other by both evaluating the graph, and
setting the equations equal to one another in terms of 𝑥, and then solve for 𝑥.
Make a copy of the graph so that there is one for the washer method, and one for the
shell method.
On one of the graphs draw in a 3-dimensional washer and identify if horizontal or vertical
cross sections are needed for the integration. Identify and label the washer’s corresponding
dimensions and solve for them.
Create and evaluate an integral which represents the volume of the 3-dimensional
structure obtained by rotating both curves about the vertical line at x = 4 using washers.
On the duplicate graph draw in a 3-dimensional shell and identify if horizontal or vertical
cross sections are needed for the integration. Identify and label the shell’s corresponding
dimensions and solve for them.
Create and evaluate an integral which represents the volume of the 3-dimensional
structure obtained by rotating both curves about the vertical line at x = 4 using shells.
Verify both answers by use of a computer algebra system (calculator).
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Data Points and Graphs
Before the washer method or shell method can be implemented a graph of the functions
must be generated. This can be achieved by evaluating both functions on the interval [0, 4], and
establishing the Cartesian coordinate data points (see figure 1 below).
Figure 1: Functions evaluated on the interval [0, 4].
f(x) =
𝑥3
𝑦=
8
X Coordinates:
0
1
2
3
Y Coordinates:
0
0.125
1
3.375
4
8
g(x) =
𝑦 = 2𝑥
X Coordinates:
Y Coordinates:
0
1
2
3
0
2
4
6
4
8
Now that the data points of both functions have been established, the next step is to plot
them on a graph. Setup an (x, y) graph, plot the corresponding data points for the functions
(color coded lines), and shade the region bounded by the curves (see figure 2 below).
Figure 2: Graph of functions before rotating about x = 4.
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Having a graph of the functions allows for the rotation of the functions about 𝑥 = 4. Use
symmetry about the vertical line 𝑥 = 4, to plot the data points on the interval [4, 8], shade in
the region bounded by the resulting curves, and draw the resulting 3-dimensional object (dashed
red lines) obtained by the rotation (see figure 3 below). It should be noted that the white area under
the f(x) (and its symmetrical counterpart) is empty or hollow.
Figure 3: Functions rotated on the interval [0, 4], about the vertical line x = 4.
The next step in setting up for both the washer method and shell method is to identify
where the functions intersect. Knowing where the functions intersect establishes the upper and
lower bounds for the integral used in the washer and shell methods.
Analyze the graph in figure 3 to see where both the f(x) and g(x) intersect one another.
The functions intersect each other at three points the first being (-4, 0), the second being
(0, 0), and the third being (4, 8). This can be proven algebraically by setting both functions equal
to each other in terms of 𝑥, isolating the 𝑥 values on one side of the equation in factored form ,
and setting each factor equal to zero solving for x (see below).
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Washer Method
The washer method uses the volume of a bolt type washer object to measure the volume
of a 3-dimensional object. The washers are fitted snugly against the inner walls of the 3dimensional object obtained by rotating f(x) and g(x) about 𝑥 = 4, with an infinitely small height,
and therefore an infinite amount of washers. The exact volume is found by taking the limit as
the amount of washers becomes infinitely large and adding all of their volumes together(e.g. see
figure 4 pg. 6).
On a duplicate graph of figure 3, draw in a 3-dimensional washer with its origin at 𝑥 = 4.
Given the following information about the volume and dimensions of a washer (see below) locate,
label, and solve for the variable dimensions of the washer drawn on the graph (see figure 4 pg. 6).
Volume of a Washer:
Figure 4 on page 6 reveals that for integration purposes changes in 𝑦 values (∆𝑦) will be
considered as the height (ℎ) of the washer. Figure 4 also reveals that 𝑅 is equal to the distance
from the origin of the 3-dimensional object being measured, minus the 𝑥 value where it meets
the g(x) at a given height (𝑦). The last variable to define is 𝑟, it is equal to the distance from the
origin of the 3-dimensional object being measured, minus the 𝑥 value of where it meets the f(x)
at a specified height (𝑦).
Taking into consideration ∆𝑦 values will be utilized using the washer method, the lower
bound for integration is 𝑎 = 0 (y value from point of intersection [0, 0]), and the upper bound is 𝑏 = 8
(y value from point of intersection [4, 8]). Both the f(x) and g(x) need to be solved in terms of 𝑦 to
define the variable dimensions of the washer.
Functions Solved for 𝑦:
Variable Dimensions Defined:
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Washer Method Calculations
Having found the variable dimensions of the washer object in figure 4, it is now time to
setup an expression of the 3-dimensional object’s volume that was obtained by rotating the
functions about the vertical line at 𝑥 = 4. Take a Riemann Sum of the volumes of the washers,
and take the limit as the number of subdivisions (number of washers) becomes infinitely large, and
by definition, a definite integral with respect to ℎ is obtained (Stewart, James). Lastly evaluate the
resulting integral from the lower bound (a = 0) to the upper bound (b = 8), and obtain an
approximation and verification of the findings with a calculator.
Figure 5: Washer Method Calculations.
8
𝒏
𝟐
𝐥𝐢𝐦 ∑ 𝝅(𝑹 − 𝒓
𝒏→∞
𝟐 )𝒉
= ∫ 𝜋(𝑅2 − 𝑟 2 ) 𝑑ℎ
𝒊=𝟏
0
Substitute variable dimensions of washer into equation.
8
= ∫ 𝜋 ([4 −
0
𝑦2
] − [4 − 2 3√𝑦]2 ) 𝑑𝑦
2
𝑦
Factor out 𝜋, simplify by expanding [4 − ]2 , and −[4 − 2 3√𝑦]2 .
2
8
2
= 𝜋∫(
0
𝑦
3
− 4𝑦 + 16 + 16 3√𝑦 − 4 √𝑦 2 − 16) 𝑑𝑦
4
Simplify by combining like terms.
8
= 𝜋∫(
0
𝑦2
3
− 4𝑦 + 16 3√𝑦 − 4 √𝑦 2 ) 𝑑𝑦
4
Take the anti-derivative of each term in the integrand.
4
5
1 𝑦3
𝑦2
3 ∙ (𝑦)3
3 ∙ (𝑦)3 8
= 𝜋[ ∙
− 4∙
+ 16 ∙
− 4∙
]
4 3
2
4
5
0
Simplify by writing fractional exponents as radicals and multiplying coefficients.
3
= 𝜋[
𝑦3
12 √𝑦 5 8
3
− 2𝑦 2 + 12 √𝑦 4 −
]
12
5
0
Evaluate F(b) – F(a) on the interval [0, 8].
(8)3
= 𝜋 ([
12
3
− 2(8)2 + 12 √(8)4 −
3
12 √(8)5
5
]–[
(0)3
12
Simplify answer.
128
384
= 𝜋 ([
− 128 + 192 −
] – [0])
3
5
The volume of the 3-dimensional object.
448𝜋
15
Verification of answer and approximation with calculator.
=
8
𝑦
448𝜋
2
15
= ∫0 𝜋 ([4 − ]2 − [4 − 2 3√𝑦]2 ) 𝑑𝑦 =
(7)
3
− 2(0)2 + 12 √(0)4 −
≈ 93.8289
3
12 √(0)5
5
])
Shell Method
The shell method uses the volume of a thin hollow circular type object with variable
dimensions to measure the volume of a 3-dimensional object. The shells are fitted snugly against
the inner walls of the 3-dimensional object obtained by rotating f(x) and g(x) about 𝑥 = 4, with
an infinite amount shells placed within the outer most shell. The exact volume is found by taking
the limit as the amount of washers becomes infinitely large and adding all of their volumes
together(e.g. see figure 6 pg. 9).
On a duplicate graph of figure 3, draw in a 3-dimensional shell with its origin at 𝑥 = 4.
Given the following information about the volume and dimensions of a shell (see below) locate,
label, and solve for the variable dimensions of the shell drawn on the graph (see figure 6 pg. 9).
Volume of a Shell:
𝑣 = 2𝜋𝑟ℎ ∆𝑥
𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑓𝑟𝑜𝑚 𝑜𝑟𝑖𝑔𝑖𝑛 𝑡𝑜 𝑜𝑢𝑡𝑒𝑟 𝑒𝑑𝑔𝑒 𝑜𝑓 𝑠ℎ𝑒𝑙𝑙.
ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠ℎ𝑒𝑙𝑙.
Figure 6 on page 9 reveals that for integration purposes changes in 𝑥 values (∆𝑥) will be
utilized using the shell method. Figure 6 also reveals that 𝑟 is equal to the distance from the
origin of the 3-dimensional object being measured, minus the 𝑥 distance from where the shell
meets both the f(x) (bottom) and g(x) (top). The last variable to define is ℎ, it is equal to the
distance between the height of g(x) minus the height of f(x), with respect to the ∆𝑥.
Taking into consideration ∆𝑥 values will be utilized for the shell method, the lower bound
for integration is 𝑎 = 0 (x value from point of intersection [0, 0]), and the upper bound is 𝑏 = 4 (x value
from point of intersection [4, 8]). Both the f(x) and g(x) need to be solved in terms of 𝑥 to define the
variable dimensions of the shell.
Functions Solved for 𝑥:
𝑔(𝑥) = 𝑦 = 2𝑥
𝑓(𝑥) = 𝑦 =
𝑥3
8
Variable Dimensions Defined:
𝑟 =4−𝑥
ℎ = 2𝑥 −
𝑥3
8
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Shell Method Calculations
Having found the variable dimensions of the shell object in figure 6, it is now time to
setup an expression of the 3-dimensional object’s volume that was obtained by rotating the
functions about the vertical line at 𝑥 = 4. Take a Riemann Sum of the volumes of the shells
stacked inside one another. Take the limit as the number of subdivisions (number of shells)
becomes infinitely large, and by definition, a definite integral with respect to 𝑥 is obtained
(Stewart, James). Evaluate the resulting integral from the lower bound (a = 0), to the upper bound
(b = 4), and obtain an approximation and verification of the findings with a calculator.
Figure 7: Shell Method Calculations.
4
𝒏
𝐥𝐢𝐦 ∑ 𝟐𝝅𝒓̅𝒉 ∆𝒙
𝒏→∞
= ∫ 2𝜋𝑟̅ ℎ 𝑑𝑥
𝒊=𝟏
0
Substitute variable dimensions of shell into equation.
4
𝑥3
) 𝑑𝑥
8
= ∫ 2𝜋(4 − 𝑥) (2𝑥 −
0
Factor out 2𝜋, simplify by Foiling (4 − 𝑥) (2𝑥 −
𝑥3
8
).
4
𝑥3
𝑥4
2
= 2𝜋 ∫ (8𝑥 − − 2𝑥 + ) 𝑑𝑥
2
8
0
Take the anti-derivative of each term in the integrand.
= 2𝜋 [8 ∙
𝑥2 1 𝑥4
𝑥3 1 𝑥5 4
− ∙
− 2∙
+ ∙ ]
2
2 4
3
8 5 0
Simplify by multiplying coefficients.
= 2𝜋 [4𝑥 2 −
𝑥4
2𝑥 3 𝑥 5 4
−
+
]
8
3
40 0
Evaluate F(b) – F(a) on the interval [0, 4].
= 2𝜋 ([4(4)2 –
(4)4
8
−
2(4)3
3
+
(4)5
40
] – [2(0)2 –
(0)4
Simplify answer.
= 2𝜋 ([64 − 32 −
128 128
+
] – [0])
3
5
The volume of the 3-dimensional object.
=
448𝜋
15
Verification of answer and approximation with calculator.
4
∫0 2𝜋(4 − 𝑥) (2𝑥 −
𝑥3
8
) 𝑑𝑥 =
(10)
448𝜋
15
≈ 93.8289
8
−
2(0)3
3
+
(0)5
40
])
Conclusions
Solving the given problem using the shell method produced a volume of exactly
[(448𝜋)/15], and an approximation of 93.8289. Using the shell method to solve the given
problem produced a volume of exactly [(448𝜋)/15], and an approximation of 93.8289.
Considering that both methods produced the same results, we have determined that the volume
of the 3-dimensional object obtained by rotating f(x) and g(x) about x = 4, is exactly [(448𝜋)/15],
and approximately 93.8289. These findings have also been verified by a computer algebra
system (calculator).
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References

Stewart, James. Calculus Concepts and Contexts, Enhanced Review Edition (with CD-ROM, Tools,
iLrn-Personal Tutor with SMARTHINKING). Belmont: Brooks Cole, 2006. Print.

"Fond ecran mario." Www.monsite.com. Monsite. Web. 6 Dec. 2009. <http://mariocape.emonsite.com/rubrique,fond-ecran,292947.html>.
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