Aquarium Calculus
Grade Level:
Duration:
Grade 11/12
6x50 min
Subject: Calculus
Prepared By: Mike Borowczak
Materials Needed
Req: Graphing Paper, Rulers. Optional: Foam Board, Graphing Calculators
Analyze Learners
Overview & Purpose (STEMcinnati theme)
Students will enhance & apply knowledge of both integration and
differentiation in calculus. Students will discover how engineers must
use calculus in creating large-scale attractions (e.g. walk-through
aquariums).
Education Standards Addressed
AP Calculus Key Concepts
II. Derivatives
E. Applications of Derivatives

Applications:
Large Scale Attractions, Design of Vehicles/Suits Under Unique
Stresses (Aircrafts, Space Vehicles, Deep Sea Explorers etc).
Careers:
Civil Engineers (and Environmental Engineers), Material Engineers,
Architects, Mechanical Engineers, Aerospace
Society:
Entertainment, Exploration


Analysis of curves, including the notions of
monotonicity and concavity.
Optimization, both absolute (global) and relative
(local) extrema.
Modeling rates of change, including related rates
problems.
III. Integrals
A. Interpretations and Properties of
Definite Integrals



Definite integral as a limit of Riemann sums.
Definite integral of the rate of change of a quantity
over an interval interpreted as the change of the
quantity over the interval:
Basic properties of definite integrals. (Examples
include additivity and linearity.)
B. Applications of Integrals
Appropriate integrals are used in a variety of applications to
model physical, biological, or economic situations. Although
only a sampling of applications can be included in any specific
course, students should be able to adapt their knowledge and
techniques to solve other similar application problems. Whatever
applications are chosen, the emphasis is on using the integral of a
rate of change to give accumulated change or using the method
of setting up an approximating Riemann sum and representing its
limit as a definite integral. To provide a common foundation,
specific applications should include finding the area of a region,
the volume of a solid with known cross sections, the average
value of a function, and the distance traveled by a particle along a
line.
Goals & Objectives
Teacher Guide
Student Guide
Assessment
Goals and
Objectives
(Specify
skills/information that
will be learned.)
Goals:
Students will understand:
1.How to construct a solid by revolving functions
around x & y axis
2.How to find the volumes of the solids listed in
#1 as well as volumes constructed by the
intersection of functions
3.How to rates of change are used in everyday
settings
4.Students will see how calculus is used by
engineers in the development of large scale
attraction & theme parks
Objectives:
1.Students will design a multi-room aquarium
using their knowledge of integration
2.Students will compute 3 volumes for each
room they design – 2 Areas below the curve, and
the Area between
3.Students will draw and construct crosssections of each room
4.Students will compute & explain the rate of
change of customers within their aquarium
during the day
1. List at least five (5) ways that
calculus applications exist in
theme park / large scale
attractions (such as Kings
Island, Sea World, The
Newport Aquarium, Cedar
Point):
2. Write an equation for
balance(t), the ratio between
the rate of visitors entering an
attraction and the rate of
visitors leaving an attraction.
Assume you are given enter(t),
exit(t), functions describing the
total number of visitors which
have entered and exited from
the attraction.
3. Using your answer for number
2, find the rate of change of
balance(t).
4. Using your answers for 2 & 3;
assume that enter(t) = 2t2+50
and
exit(t) = 25 –(t-5)2 find
balance(t) and the rate of
change of balance(t):
Balance(t) =
Balance’(t) =
5. Given a building with width and
length of 94 ft, and height 30 ft,
a) Find the Volume of the
building.
b) Assume that we fill the
Select Instructional
Strategies –
Information
(Catch, give and/or
demonstrate necessary
information,
misconceptions, etc…)
Utilize Technology
Day 1
Pre-assessment (10 minutes)
 Questions to the right
Catch (n = 10 min)
 Discussion of Local Theme Parks
 Leading Questions: how do they
design:
o coaster loops so that people
don’t fall
o tunnels so that a Million
Gallons of water don’t
implode on visitors
o how large a concession area
is needed in the middle of an
attraction
Lesson (continuous during activity)
o Reinforce integration of
volumes and solids around
axis
o Reinforcement of Rates-ofChange
Discussion (n = 5 min)
 Discuss Key Observations
Homework (Student Determined)
 Students may work on their
projects outside of school hours
(e.g. added features, visual appeal)
 Wolfram Alpha can be used for free
visualization
Catch
Students should notice that
calculus is required to
perform computations that
would otherwise would be
determined by trial and error.
Other Resources
(e.g. Web, books,
etc.)
Require Learner
Participation
Activity
(Describe the
independent activity to
reinforce this lesson)
Day 2-5
Overview (n = 5 minutes)
 Handout Aquarium Calculus Instructions
 Students Paired
 Review Expectations & Deliverables
Activity (4 days)
Overview
The XYZ Economic Development Council has determined, using
market analysis and surveys, that in order to bring in local revenue
and long-term jobs to the area – the creation of a large-scale
attraction style aquarium is a viable investment project.
Request
The council is soliciting initial proposal bids from a minimum of 5
Engineering & Design firms. The top 3 bids will be partially funded
for further analysis & forwarded to the architectural bidding phase.
Constraints
1][An area of land has already been set aside for the project, and
due to zoning regulations, the maximum size of the building is set
at 94 ft wide, 94 ft long and 30 ft tall.
2][In order to be competitive with other comparable local
attractions (The Newport Aquarium) your bid must include an
aquarium with contains at least 1 Million Gallons of water
(preferable more).
Write the function the shows the amount of water in your aquarium
water = total volume of building – volume of aquarium rooms
3][You must design a minimum of 4 separate rooms. The rooms
must have the following characteristics:
The rooms must be formed by rotating a 2-D shape around a line
At least 3 of the rooms must be created by the intersection of two
lines
4][You must draw out the three views of you aquarium layout to
scale
Sagittal View (Y-Plane)
Coronal View (X-Plane)
Transversal View (Z-Plane)
5][Each member must each also create the cross-sections for at
Evaluate
(Assessment)
Day 5
Standard Post-Assessment
Additional Notes
(Steps to check for
student understanding)
– See Objectives above
Important Attachments:
1. Pre-Post Assessment
2. Worksheets
3. PowerPoint
4. Reflection after lesson
Reflection:
What worked:
Student’s enjoyed a hands on application to calculus – e.g. sketching an aquarium, defining equations to form the walls,
transferring the forms to foam and cutting it out. Concept to Implementations they could see how calculus gets used to
engineers & designers. Students also enjoyed the re-enforcement of seeing a large scale attraction and being able to
compute the volume & surface areas of not only the aquarium walls, but the visitor occupation volume.
What needs more polishing:
Student groups were split on designing their tanks – some groups wanted more direction – they didn’t understand that it
was freeform- they were too worried about the calculus than enjoying the fact that they were free to create anything. In
order to combat this a prize mechanism – e.g. group able to have the closest balance between water volume and guest
volume get prize XYZ, the group with the most complex shape get ABC, and so forth.
Additionally, as expected, this activity brought out fundamental misunderstands on the students part for computation of
volumes by revolving areas. As such, our timing slipped and we ended up taking 7 classes + 1 day for a field trip.