Extra Credit Assignment – Math 201
Complete ONE of the following tasks:
1) Construct a volume using appropriate materials that models a given situation. Materials
could include pipe cleaners, cardboard, clay, straws, etc. Compute its volume and surface
area as well. Turn in the model and calculations of the volume and surface area.
2) Construct a centroid mobile using appropriate materials, such as graph paper glued to
cardboard, cardstock, or other hard material. Compute the centroid of each shape chosen.
Turn in the mobile and calculations used to find the centroid of chosen shapes.
**Up to 10 points toward a test will be awarded for a quality construction with calculations.
You will be evaluated on accuracy, quality, clarity, and overall aesthetics. You must have a
model to get credit. This assignment is due Monday, June 2.
Instructions for Task 1: Constructing a volume.
Some volumes are created using geometric descriptions over an area. For example, consider a
volume whose base is a circle, but with cross-sectional heights that are squares. (See below)
View from corner
Bottom View
Side (facing squares)
Side (side of squares)
Another volume created in this manner could be an object with an equilateral triangle base, with
cross-sections that are hemispheres. (See 6.2 #52)
Your task is to construct ONE of the following volumes and calculate its volume and surface
area. You will hand in the volume constructed as well as the calculations for finding the volume
and surface area. Calculations should include the setup of the integral, but technology may be
used to evaluate the integral.
Object Choices:
1) Circular base with cross-sections that are equilateral triangles
2) Semicircle with equilateral triangle cross-sections perpendicular to the line of the semicircle
3) Base bounded by y = x and y = 4 − x with semicircular cross-sections parallel to the x-axis
4) Base bounded by y = x 2 and y = 4 − x 2 with square cross-sections parallel to x-axis
5) Challenge: Base bounded by r = 2 cos ( 2θ ) with square cross-sections perpendicular to each
axis petal.
Instructions for Task 2: Constructing a centroid mobile.
Your task is to construct a mobile with 4 shapes and 3 rods so that it looks like below, where
shapes lay flat and are attached to the mobile at their centroids.
You may use any (stiff) material you like such as cardboard, foam, etc. for the objects and
dowels, sticks, etc. for the rods. Paperclips and string are commonly used for attaching the
objects to the dowels.
You will need to carefully draw the shapes on graph paper, then cut out the corresponding shapes
for the mobile from the stiffer material you chose. Graphs used may be glued (or taped) to the
final product, but this is not required. Graphs should be turned in with the mobile if they are not
adhered to the object shapes. You should also turn in a sheet of paper containing the integral
setup for each centroid, although technology may be used to evaluate the integrals.
Centroid object choices:
1) Region bounded by f ( x ) = x 2 − x + 3 and g ( x ) = − x + 1 on the interval [ −1,3] .
2) Region bounded by f ( x ) = x 2 − 3 x + 5 and g ( x ) = 1 on the interval [ −1,3] .
3) Region bounded by f ( x ) = 2 x − 4 and g ( x ) = 2 x − 12 on the interval [ 0, 4] .
4) Region bounded by f ( x ) = 3 x and g ( x ) =
x2
.
9
5) Region bounded by f ( x ) = x 2 − 3x − 8 and g ( x ) = − x .
6) Region bounded by f ( x ) = e x and the x-axis on the interval [ 0,3] .
7) Region bounded by f ( x ) = − x + 11 and g ( x ) =
10
.
x
8) Region bounded by f ( x ) = 2 x + 4 and g ( x ) = x 2 − 4 x + 4 .
9) Region bounded by f ( x ) = 3e x and the x-axis on the interval [ −2, 2] .