The Prolate Spheroid
A Prolate Spheroid is a solid of revolution formed by rotating an ellipse about its major
(longer) axis, resulting in a shape elongated along this axis, like a rugby ball or American
football. This contrasts with an oblate spheroid, which is flattened and formed by rotating an
ellipse about its minor axis. The Prolate Spheroid has two equal equatorial radii and a longer
polar radius, giving it a "pointy" elongated shape.
Brief Discussion/Introduction
This project focuses on using the method of solid revolution to find the volume of a
3D solid formed by rotating the region under the curve y = 2sinx around the y - axis. By using
the shell method, we were able to compute the volume efficiently.
Understanding methods like the shell and disk methods is important in solving real-
life problems involving volume, such as determining the capacity of tanks, bottles, or
sculptural forms.
The solid generated by this rotation closely resembles an American football. It has a
smooth, symmetrical shape that bulges in the center and tapers at both ends — just like the
shape formed by rotating the sine curve. This demonstrates how mathematical modeling
and calculus are used to describe and calculate volumes of real-world objects
Understanding methods like the shell and disk methods is not only essential in
calculus but also highly practical in real-life applications. These techniques allow us to find
the exact volume of irregular or curved solids that cannot be measured easily using basic
geometry. For example, in our activity, the 3D shape formed by rotating the curve y = 2sinx
about the y-axis resembles the shape of an American football. This demonstrates how
calculus can model real-world objects, such as sports equipment, bottles, machine parts, or
even artistic sculptures.
Choosing the right method—whether shell or disk—can simplify complex calculations,
especially when one method avoids difficult integrals. This shows the importance of having
multiple approaches in mathematics, enabling us to solve problems more efficiently and
accurately. Overall, mastering these methods enhances both mathematical thinking and
practical problem-solving in engineering, design, manufacturing, and other fields.
2D Graph Illustration
Generated Solid
Sides and Top View
TOP VIEW BACK LEFT SIDE RIGHT SIDE
Volume using Disk/Washer Method
State the problem
Find the volume of the given solid.
Answer: According to the given function: (y= 2sinx and x - axis, axis of revolution: y - axis)
It leads to a difficult integral with inverse trigonometric squared and it requires tricky
substitution or numerical methods. The integral doesn’t simplify nicely, so its very hard (or
impossible) to solve by hand.
Graph and Labels
Volume of the Solid using Shell Method
State the problem:
Find the volume of the solid.
Graph and Labels
Solution: :
π
∫
V = 2π x (2sinx)dy
0
∫
= 4π x sin x dy
0
[ ]
π
∫
= 4π −x cos x |π0 + cos x dx
0
= 4π[π + sin x|π0]
= 4π[π + o]
= 4π(π) ≈ 39.478
Conclusion: Therefore, the volume of the Prolate Spheroid using the shell method is
approximately 39.478 cubic units.
